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3.1 Background to the Polygonal Technique

Artistic methodology is an important aspect of any serious study of art. A detailed knowledge of the methods and techniques used by traditional artists, craftsmen, and architects will provide a more complete understanding of the bold trends, as well as subtle nuances within a given artistic discipline. As regards Islamic geometric patterns, the understanding of historical design methodology provides valuable insight into the initial development, refinement and maturation, and geographic distribution of this design tradition. However, such knowledge is not just relevant to historians of Islamic art and architecture. A detailed understanding of historical design methodology is especially germane to those who are involved with the incorporation of such patterns into their own creative enterprises. Yet, with the historical decline of this ornamental tradition, knowledge of the methods used to produce complex patterns was gradually lost. Even with the resurgence of interest in Islamic geometric patterns that began during the second half of the twentieth century, attempts to resuscitate this art form have been stymied due to the lack of understanding of historical methodology. This ongoing void has caused frustration among contemporary artists, designers, and architects who have had to rely upon merely copying existing designs. The loss of vitality is a great pity, for this design discipline still has much to offer. Indeed, the potential for new and original geometric patterns draws from an infinite pool that can never run dry. New designs are there for the creation, and a practical knowledge of this design methodology can be a great inspirational aid and indispensable tool for those engaged in the revival of this extraordinary artistic tradition.

In light of the above, the detailed methodological analysis provided in this chapter serves two functions: to better understand the rich diversity of historical Islamic geometric designs and to provide artists and designers with familiarity of the technical skills required for creating new and original patterns at even the most demanding levels of geometric complexity. Much can be learned from the methodological practices of the past, and there is tremendous scope for new discoveries that augment the exceptional work of past masters working within this diverse discipline.

The historical evidence for the polygonal technique establishes this as the preeminent design methodology used by Muslim artists throughout the long history of this tradition. It is therefore no surprise that the polygonal technique is especially appropriate for creating new geometric designs that conform to long-established Muslim aesthetics. The succeeding methodological analyses in this chapter examine the application of the polygonal technique to the full range of pattern types employed within this ornamental tradition. This includes each of the five historical systematic methodologies, as well as the diversity of nonsystematic geometric design varieties. Additive and subtractive variations are provided where relevant to historical examples. What is more, the specialized techniques used in creating historical dual-level patterns are extended to fulfill the precise qualifiers for self-similarity and true quasicrystallinity. This chapter concludes with an examination of the two historical varieties of domical geometric pattern application: those that employ gore segments as their repetitive device and those based upon polyhedral geometry such as the Platonic and Archimedean solids. Interspersed throughout this chapter are a small number of designs created by the author. Individually, these serve to highlight the further design potential of a given underlying polygonal tessellation, or repetitive stratagem. Collectively, these serve to touch upon the vast potential of the polygonal technique for creating new and original designs that fully conform to, or in some cases build upon, the aesthetic character of this historical art form.

The working practices associated with applying geometric patterns to the wide range of ornamental media—be it architectural, the book arts, or otherwise—are governed by their own set of practical requirements and cultural conventions. Such considerations are highly specific and generally beyond the scope of this current work. However, the conventions for panelizing, or framing, geometric patterns are fundamental to their applied use, regardless of artistic medium. The repetitive grids that are fundamental to the ability of patterns to cover the two-dimensional plane provide a wide range of proportional choices for applying designs into bounding frames. Figure 84 illustrates a number of typical framing rectangles created from repetitive grids that are proportioned upon repeat units common to Islamic geometric patterns. Any of these can be extended or reduced incrementally, and the vertical and horizontal lines of the frame generally conform to lines of symmetry within the geometric pattern.

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Fig. 84

Another secondary methodological practice involves the treatment of the pattern lines within a given design. Depending on such variables as line thickness, interweave, or use of double lines, a given design can have many contrasting aesthetic qualities. Figure 85 demonstrates diverse pattern line treatments applied to the classic fivefold acute pattern. Standard treatments include widened lines of variable thickness; interweaving lines of variable thickness; various forms of double-line treatments; and basic pattern with simple color differentiation applied to the pattern’s cells (tiling treatment). Figure 86 illustrates the same forms of pattern line treatment applied to the classic fivefold obtuse pattern; Fig. 87 shows these same treatments to the pattern lines of the classic fivefold median design; and Fig. 88 provides typical pattern line treatments to the classic fivefold two-point pattern. The choice of which variety of pattern line treatment to use in a given historical example would have been determined by the aesthetic predilections of the artist as influenced to a greater or lesser degree by inherent geometric conditions, cultural aesthetic conventions, and material constraints of the designated medium.

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Fig. 85

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Fig. 86

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Fig. 87

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Fig. 88

3.1.1 Systematic Design: System of Regular Polygons

As discussed previously, the use of polygonal systems to create Islamic geometric patterns can be traced back to the formative period of this ornamental tradition, and the earliest system to be widely employed was the system of regular polygons. This makes use of regular triangles, squares, hexagons, and dodecagons as repetitive modules upon which pattern lines are applied. The variety of historical designs that can be created from tessellations comprised of different combinations of these polygons is surprisingly large. The most basic are of course designs that are derived from the three regular tessellations: the triangular grid, square grid, and hexagonal grid [Fig. 1]. More visually compelling and geometrically interesting patterns are created from the semi-regular, two-uniform, and three-uniform tessellations made up of these repetitive modules. Figure 89 shows the eight semi-regular tessellations. These are characterized by a single variety of vertex with more than a single variety of polygon. The repetitive structures of five of these tessellations are isometric; two are orthogonal; and one repeats with elongated hexagons. Figure 90 illustrates 12 examples of two-uniform tessellations. These are characterized by two varieties of vertex. There has been some disagreement over the precise number of two-uniform tessellations. Depending on whether the topologies of the two varieties of vertex have consistent or inconsistent global settings, the number of two-uniform tessellations can be limited or unlimited respectively.Footnote 1 The examples shown have either isometric, orthogonal, rectangular, or rhombic repeat units. Figure 91 illustrates just 4 three-uniform tessellations. This type of tessellation has three types of vertex. The repetitive structures of these particular examples are either isometric or orthogonal. Geometric designs that are created from semi-regular, two-uniform, and three-uniform tessellations will invariably adhere to the same repetitive structure as the generative tessellation.

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Fig. 89

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Fig. 90

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Fig. 91

Within the historical record there is great diversity in the pattern line application to the repetitive modules that comprise the system of regular polygons. The chart in Fig. 92 provides the more typical pattern line applications to the triangle, square, hexagon, and dodecagon. The octagon has been excluded from this chart due to the fact that it will only produce one tessellation with the other regular polygons: the 4.82 semi-regular tessellation of squares and octagons. Due to this limitation, and for the purposes of this work, the many patterns created from this tessellation are treated as a special case, and excluded from the system of regular polygons. The chart in Fig. 92 identifies the various pattern line applications as being acute, median, obtuse, or two-point. However, unlike the other historical design systems, or indeed the conventions for creating nonsystematic patterns, there are more that just four primary forms of pattern line application in the system of regular polygons. It is therefore helpful to further differentiate the types of pattern line by their angles. In this way, there are two types of median and obtuse pattern, and five types of two-point pattern. Figure 93 demonstrates five patterns created from different construction sequences as applied to the same 32.4.3.4-3.4.6.4 two-uniform tessellation. Each of these begins by initially populating either the triangles and hexagons or just the squares with a specific set of pattern lines and extending these into the adjacent polygonal cells until they meet with other extended lines. Figure 93b illustrates two patterns that can be constructed from the simple application of squares within the square modules. Each of these five patterns is by the author, and is not known to have been used historically. Figure 94 illustrates the same process as applied to the triangle and square cells of the 32.4.12-3.4.3.12-3.122 three-uniform tessellation. Again, each of these four patterns is by the author and are not known to the historical record, but, like the previous five examples, is within the aesthetic scope of traditional Islamic design. Keeping in mind the innumerable tessellations that can be created from the regular polygons, the examples from Figs. 93 and 94 demonstrate the vast potential for new and original patterns that are still available to the system of regular polygons.

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Fig. 92

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Fig. 93

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Fig. 94

As mentioned, patterns can be created by applying pattern lines to the polygonal cells of the three regular grids. Historically speaking, this is especially true of the 63 tessellation of regular hexagons. Figure 95 illustrates a series of patterns created from this simple tessellation. Figure 95a shows an acute pattern with 30° crossing pattern lines. This was used by Seljuk artists within the northeast dome chamber of the Friday Mosque at Isfahan (1088-89) [Photograph 18], and the Friday Mosque at Sin in Iran (1134), as well as by Fatimid artists at the Sayyid Ruqayya Mashhad in Cairo (1133). Figure 95b shows the classic threefold median pattern with 60° crossing pattern lines. This is one of the most widely used geometric patterns throughout Muslim cultures, and two fine examples include: a pair of wooden doors at the Aljafería Palace in Zaragoza, Spain (second half of the eleventh century), and the Mamluk window grilles of the Sultan Qala’un funerary complex in Cairo (1284-85) [Photograph 55]. Figure 95c is a median pattern with 90° crossing pattern lines. This design is similarly ubiquitous, and a particularly impactful example from the Seljuk Sultanate of Rum is found at the Sultan Han near Aksaray (1229). Figures 95d and e show obtuse patterns with 120° crossing pattern lines. The linear bands of Fig. 95d are, in and of themselves, the 3.6.3.6 tessellation with a widened interweaving interpretation. This design is ubiquitous throughout Muslim cultures. Figure 95e is by the author, and not known to have been used historically. Figure 95f employs pattern lines that connect to the vertices of the hexagonal grid. This design can also be produced with 120° obtuse pattern lines (dashed lines) that are widened to their maximum extent. This too was popularly used throughout Muslim cultures. Figure 96 illustrates 9 two-point patterns that are easily created from the 63 tessellation. Fig. 96a–f all use more typical pattern line applications, and Fig. 96g–i employ less common pattern line applications. The pattern in Fig. 96a is surprisingly uncommon and appears to have been principally used in Persian miniature painting. Figure 96b was used ubiquitously, and a relatively early example from the Seljuk Sultanate of Rum is from the Alaeddin mosque in Kırşehir, Turkey (1230). The very well known design in Fig. 96c was used at the Sabz Pushan outside Nishapur (960-85) and originates during the period of Samanid influence over this region [Photograph 11]. This design was also used on the eastern tomb tower at Kharraqan (1067-68). The pattern of superimposed hexagons in Fig. 96d was also widely used, and early examples include an Umayyad marble grill from al-Andalus (tenth century), and the Ghurid raised brick ornament at the Friday Mosque at Herat (1200). The design in Fig. 96e was equally well used and early examples include: a Fatimid window grill at the al-Azhar mosque in Cairo (970-76); an Umayyad window grille at the Great Mosque of Córdoba (987-990); two Seljuk examples from the eastern tomb tower at Kharraqan (1067-68) [Photograph 17] and the Friday Mosque at Abyaneh, Iran (1073); and a Fatimid example from the al-Aqmar mosque in Cairo (1125). The Qarakhanid design in Fig. 96f originates from the Maghak-i Attari mosque in Bukhara, Uzbekistan (1178-79), and has the further distinction of incorporating the generative hexagonal grid into the finished design. Figure 96g shows a Saminid design from the mausoleum of Arab Ata at Tim, Uzbekistan (977-78) [Photograph 12]. The earliest known use of the closely related pattern in Fig. 96h is from the eastern tomb tower at Kharraqan (1067-68). The design in Fig. 96i is Mengujekid from the Great Mosque of Divrigi in Turkey (1228-29). The three patterns in Fig. 97 are less typical two-point patterns associated with this same 63 tessellation that incorporate higher order polygons into the pattern matrix. The pattern in Fig. 97a was used by artists working for the Seljuk Sultanate of Rum for the Ahi Serafettin mosque in Ankara (1289-90). This can also be created from an alternative tessellation that employs underlying ditrigonal shield modules [Fig. 118c]. Several historical patterns were produced that are essentially variations on this design, including an early Seljuk example [Fig. 118a] from the northeast dome chamber of the Friday Mosque at Isfahan (1088-89) [Photograph 19]. When using the 63 tessellation to create the design in Fig. 97a, octagons are placed upon the midpoints of each edge of the generative hexagonal grid, with two corners of each octagon falling upon the polygonal edge. In this construction, the size of the octagon determines the character of the finished design. The design in Fig. 97b was used in multiple locations during the Seljuk Sultanate of Rum, including the Great Mosque at Bayburt (1220-35), the Çifte Minare madrasa in Erzurum (later thirteenth century), and at the Ahi Serafettin mosque in Ankara (1289-90). This design locates nonagons at the vertices of the generative hexagonal grid, and uses the grid itself within the completed pattern. Figure 97c represents a relatively common design that is comprised of a matrix of superimposed dodecagons that are located on the vertices of the isometric dual grid with the dodecagonal corners placed upon two points of each hexagonal edge. There are several nearly identical historical designs that employ superimposed dodecagons within an isometric repetitive structure, each created from a different underlying polygonal tessellation. In addition to the regular hexagonal grid of Fig. 97c, the 3.6.3.6 tessellation will also create a version of this design [Fig. 99b], as will the 3.4.6.4 tessellation [Fig. 107d]. These three design variations only differ in the size of the dodecagons relative to the isometric repeat. The particular proportions of the example from Fig. 97c conform to a Fatimid window grille at the al-Azhar mosque in Cairo (970-72), as well as to a Seljuk carved stucco panel from the Friday Mosque at Forumad in Iran (twelfth century).

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Fig. 95

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Fig. 96

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Fig. 97

A number of historical patterns were created from the 63 tessellation that employ two varieties of pattern line application into adjacent hexagonal cells. Figure 98 illustrates four such examples from the historical record. The blue hexagons in this figure contain standard pattern line applications, whereas their surrounding hexagons have pattern lines that are either pattern line extensions from the blue cells, or arbitrary additions within the remaining hexagonal structure. The use of contrasting pattern lines within adjacent cells is relatively unusual, and generally dates from the formative period of this methodological tradition. Figure 98a was used by Ghurid artists in the decoration of the Masjid-i Jami in Herat, Afghanistan (1200); Fig. 98b is a later Ottoman example from the Great Mosque of Bursa (1396-1400) that is closely related to Fig. 98a; Fig. 98c shows a Seljuk design from the western tower at Kharraqan (1093-94); and Fig. 98d shows a Fatimid design from the minbar of the Haram al-Ibrahimi in Hebron, Palestine. The pattern in Fig. 98d can just as easily be created from either the 3.4.6.4 semi-regular tessellation [Fig. 106c], or the 34.6-33.42-32.4.3.4 three-uniform tessellation [Fig. 114a]. As is often the case, it is not possible to know for certain which generative structure was used to produce a given historical example.

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Fig. 98

Figure 99 illustrates several historical designs that can easily be created from the 3.6.3.6 semi-regular tessellation. Figure 99a places 60° crossing pattern lines at the midpoints of each polygonal edge of the generative tessellation, and a very similar two-point design can be created from the 63 tessellation in Fig. 96e; Fig. 99b places 90° crossing pattern lines at the same locations; and Fig. 99c places 120° crossing pattern lines at these midpoints. Figures 99d–f show different varieties of two-point patterns that locate the pattern lines at two points on each polygonal edge. It is interesting to note the similarity between the patterns in Fig. 99c and f. Both place hexagons within the underlying triangular cells, and their differences result from the pattern lines that are chosen to penetrate into adjacent hexagonal cells. The design in Fig. 99a was frequently used throughout Muslim cultures, and early examples include an Umayyad window grille from the Great Mosque at Córdoba in Spain (987-99), and a wooden top rail in the Seljuk minbar of the Friday Mosque at Abyaneh, Iran (1073). The pattern in Fig. 99b is also well known, with differing proportions resulting from different polygonal extractions (e.g. Fig. 97c). An early Seljuk example of this particular variation is found at the Friday Mosque at Golpayegan in Iran (1105-18). A particularly beautiful Ilkhanid example of the well known design in Fig. 99c was used in a frontispiece of the 30-volume Quran (1313) commissioned by Sultan Uljaytu and calligraphed and illuminated by ‘Abd Allah ibn Muhammad al-Hamadani.Footnote 2 The less common two-point design in Fig. 99d was used by atabeg artists on the sarcophagus in the mausoleum of Sultan Duqaq in Damascus (1095-1104), and in a Ghurid mihrab at Lashkar-i Bazar (after 1149) [Photograph 31]. The two-point pattern in Fig. 99e is relatively common, and can be regarded as a variation of the design in Fig. 99a, but with different proportions within the geometric matrix. A fine Mudéjar example of this design is found in the raised brick ornament on the north side of the Cathedral of San Salvador at the Aljafería Palace in Zaragoza, Spain. Each of these five examples is characterized by the superimposition of a single closed polygonal motif. By contrast, the two-point design in Fig. 99f is comprised of a network of meandering lines that do not loop back onto themselves to close a geometric circuit. This dynamic design was used during the Seljuk Sultanate of Rum at the Ali Tusin tomb tower in Tokat, Turkey (1233-34), and in the window grilles of the Sultan Qala’un funerary complex in Cairo (1284-85) [Photograph 55]. The four designs in Fig. 100 are likewise associated with the 3.6.3.6 tessellation of triangles and hexagons. Figure 100a is an interlocking pattern that begins with six-pointed stars placed at the midpoints of the hexagonal cells, but joins these stars with single lines that connect the points (rather than the more conventional extension of the pattern lines into adjacent cells as per Fig. 99a). The rotational quality of the trilobed motif breaks symmetry with the underlying polygonal structure and distinguishes this design as conforming to the p6 plane symmetry group, whereas the symmetry of the 3.6.3.6 tessellation on its own is p3m1. This is unusual in that geometric patterns generally adhere to the same plane symmetry group as their underlying generative tessellation. An early example of this design from the Seljuk Sultanate of Rum is found at the Great Mosque of Siirt, Turkey (1129), and a later Mughal example is from the tomb of I’timad ad-Dawla in Agra (c. 1628-30). Figure 100b is a two-point pattern with the applied pattern lines placed perpendicular to the edge. This example has the additional feature of including the generative polygonal tessellation as part of the completed pattern. The widening of the applied pattern lines follows the standard practice of equal offsets in both directions, while the widening of the hexagons within the generative tessellation is only in a single direction. This is unusual and highly effective in this circumstance: producing more evenly sized background elements than would otherwise be the case. This design was by an Armenian Christian artist on a stone khachkar (fourteenth century), and is also found at the Khoja Khanate ornament of the Apak Khoja mausoleum in Kashi, China (c. seventeenth century). Figure 100c places nonagons at the centers of each generative triangle that are sized so that two of their vertices fall upon each edge of the generative triangles. This design has several historical locations, including: a thin border at the Seljuk Gunbad-i ‘Alaviyan in Hamadan, Iran (late twelfth century) [Photograph 22]; the Mengujekid ornament of the Great Mosque of Divrigi, Turkey (1228-29); the Seljuk Sultanate of Rum madrasa of Muzaffar Barucirdi in Sivas, Tirkey (1271-72); and the Mamluk door of the Vizier al-Salih Tala’i mosque (1303). Figure 100d illustrates another two-point pattern. The pattern lines of this unusual design are laid out with squares placed at each vertex of the generative tessellation, and additive six-pointed stars within each hexagonal cell. As with Fig. 100b, this design also incorporates the generative tessellation as expressed by hexagons that touch corner-to-corner. This design was used in the Mamluk madrasa of Aqbughawiyya (1340) at the al-Azhar mosque in Cairo.

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Fig. 99

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Fig. 100

Figure 101 illustrates four historical examples of 3.6.3.6 patterns that employ both active and passive underlying polygonal cells in their creation. The active hexagonal cells in these examples are blue, and are separated by a passive hexagonal cell in each of the three orientations. The orange triangles are likewise active, and the white triangles are passive. It is interesting to note that each of the four designs places the same pattern lines within the active triangles, whereas each of the active hexagons is different. Figure 101a extends the pattern lines contained within the active hexagons and triangles until they meet within the passive hexagons. This design was used by Seljuk artists in the entry portal of the Seh Gunbad tomb tower in Orumiyeh, Iran (1180), as well as by Seljuk Sultanate of Rum artists at the Great Mosque of Niksar, Turkey (1145). A variation of this design was used as a border at the Çifte Minare madrasa Sivas, Turkey (1271) [Photograph 41]. Figure 101b is more conveniently created from the 34.6-33.42-32.4.3.4 three-uniform tessellation of triangles, squares, and hexagons [Fig. 114c]. This Fatimid design is found at the Sayyid Ruqayya Mashhad in Cairo (1133). The design in Fig. 101c was used during the twelfth century by Zangid artist at the Bimaristan Arghun in Aleppo [Photograph 36], during the Seljuk Sutanate of Rum at the Great Mosque at Niksar, Turkey (1145), and by the Ildegizids at the mausoleum of Yusuf ibn Kathir in Nakhichevan (1161-62). Two later Anatolian examples are from the Alaeddin mosque in Konya (c. 1220), and the Huand Hatun Complex in Kayseri (1237). The fact that the designs from Figs. 101a and c were both used in the same building in Niksar, and that both of these patterns are created from the same underlying tessellation, is an indirect source of evidence for the use of the polygonal technique within this tradition. The closely related Ildegizid design in Fig. 101d is from the Mu’mine Khatun mausoleum in Nakhichevan, Azerbaijan (1186).

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Fig. 101

Figure 102 illustrates two patterns created from an alternative arrangement of active and passive cells from the same 3.6.3.6 generative tessellation. In this case, the isometric arrangement of the primary hexagonal cells (blue) is separated by two triangles and one centrally located secondary hexagon (grey). Figure 102a shows two historical treatments for the first of these designs: each with a different widened line thickness. This pattern is generated from the placement of 60° crossing pattern lines at the midpoints of the primary hexagonal edges. Figure 102a1 shows a design from a Ghaznavid stone relief panel from the South Palace at Lashkar-i Bazar in Afghanistan (before 1036) [Photograph 13], and the example in Fig. 102a3 is from the doors of the Zangid minbar at the al-Aqsa mosque in JerusalemFootnote 3 (1168-74). Figure 102b1 places 90° crossing pattern lines at the midpoints of the active hexagonal edges, and Fig. 102b2 arbitrarily adds dodecagons into the pattern matrix, thus creating a composition comprised of superimposed dodecagons and distinctive ditrigonal shield shapes. This design was used during the Mamluk period at the private house of Zaynab Khatun Manzil in Cairo (1468).

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Fig. 102

Figure 103 shows a pattern created from the 3.3.4.3.4 semi-regular tessellation of double triangles and oscillating squares. The size of the octagons located at the vertices of the generative tessellation is determined by their extended lines bisecting the midpoints of each edge of the underlying square modules. This is a Khwarizmshahid design from the Zuzan madrasa (1219) in northeaster Iran [Photograph 39].

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Fig. 103

Figure 104 illustrates four relatively simple designs created from the 3.4.6.4 semi-regular tessellation. Figure 104a employs 60° crossing pattern lines placed at the midpoints of the underlying tessellation. This is a Mamluk median pattern used at the Aydumur al-Bahlawan funerary complex in Cairo (1364). Figure 104b places an octagon within each underlying square module and consequently has 135° crossing pattern lines at the midpoints of each underlying polygonal edge. This obtuse design was used during the Seljuk Sultanate of Rum at the Gök madrasa and mosque in Amasya, Turkey (1266-67), as well as during the Mamluk period at the Sultan Qansuh al-Ghuri Complex in Cairo (1503-05). Figures 104c and d employ hexagons within the underlying triangles. Figure 104c is an unusual example of a hybrid median and two-point pattern wherein three sets of the pattern lines contained within the triangles (the 120° median pattern lines) are extended until they meet the edges of the underlying hexagonal modules, at which point the design becomes two-point. This layout of the applied pattern lines results in a design comprised of superimposed elongated hexagons. This is a Ghurid design from the Shah-i Mashhad in Gargistan, Afghanistan (1176). The example in Fig. 104d extends the alternative three sets of pattern lines within each underlying triangle. This pattern includes an arbitrary pattern line treatment within the underlying squares, and the six-pointed star motif is placed atypically on the corners of the underlying hexagons. The overall 3.4.6.4 generative tessellation is expressed within the completed design through emphasizing just the square module. This inventive design was used by the Qarakhanids in the anonymous southern tomb in Uzgen (1186), as well as during the Seljuk Sultanate of Rum at the Izzeddin Keykavus hospital and mausoleum in Sivas (1217-18). The atypical nature of the pattern line application of this design suggests that the former of these two historical examples was a direct influence upon the latter. Figure 105 demonstrates 9 two-point patterns created from the 3.4.6.4 semi-regular tessellation. The location of the pattern lines that bisect the underlying polygonal edges in Fig. 105a, b, d, g–i is determined by the 120° pattern lines of the small hexagons placed within the underlying triangular modules. The design in Fig. 105a was used by the Ghurids at the western mausoleum at Chisht, Afghanistan (1167), and by the Seljuks in the Friday Mosque at Gonabad in Iran (1212). Fig. 105b includes six-pointed stars placed at the corners of the underlying hexagons. This design is from the Qarakhanid anonymous southern tomb in Uzgen (1186). Figure 105c includes the hexagons from the underlying tessellation within the completed design. This design was used in several locations, including the Ghurid minaret of Jam (1174-75 or 1194-95), and the Chaghatayid mausoleum of Tughluq Temür in Almaliq in western China (1363). The very similar design in Fig. 105d was also in this same building [Photograph 70]. The pattern in Fig. 105e is created by placing eight-pointed stars within each underlying square module. This design was depicted for use in a cast metal door in the Book on the Knowledge of Ingenious Mechanical Devices by Ismail ibn al-Razzaz al-JazariFootnote 4 (1206). Figure 105f (by author) is a variation of Fig. 105e that mirrors every other point of the eight-pointed stars such that they become crosses—thereby creating nine-pointed stars centered on the triangular modules. Figures 105g–i are similar in that they incorporate 12-pointed stars within the underlying hexagonal modules. 12-pointed stars are typically derived from an underlying dodecagon, and their construction in these three examples is unusual. Figure 105g was used during the Seljuk Sultanate of Rum on the façade of the Usta Sagirt tomb in Ahlat, Turkey (1273); Fig. 105h shows a design from the Mudéjar zillij mosaic ornament at the Alcázar in Seville (1362), as well as the carved plaster ornament in the synagogue in Córdoba, Spain (1315); and Fig. 105i shows a design from the Huseyin Timur tomb in Ahlat, Turkey (1279). The widening of the pattern lines in Figs. 105a, b, e, f is offset in both directions as per standard convention, while that of Figs. 105c, d, g–i is offset in only a single direction, thereby providing a better overall balance in the size of the background elements. Figure 106 illustrates three further examples of patterns created from the 3.4.6.4 semi-regular tessellation that, to a greater or lesser extent, express the generative tessellation within the completed design. Figure 106a shows a two-point pattern with the generative tessellation represented as interweaving superimposed dodecagons. This was used during the Seljuk Sultanate of Rum at the Izzeddin Keykavus hospital and mausoleum in Sivas, Turkey (1217-18). Figure 106b shows a two-point pattern that includes the extended edges of the generative triangles to produce the conditions that provide for the 12-pointed stars. This is a Fatimid example from the al-Amri mosque in Qus, Egypt (1156). Figure 106c extends the edges of the generative hexagons until they meet inside the squares. The single line six-pointed star rosettes within the underlying hexagons are additive elements. This is also a Fatimid design from the al-Amri mosque in Qus. Figure 107 shows four patterns created from the 3.4.6.4 semi-regular tessellation that are made up of superimposed higher order polygons. Such designs have their own distinctive aesthetic. Figure 107a places octagons on the center point of the generative square modules. The size of the octagons is determined by their bisecting the midpoints of the generative triangles, thereby creating ditrigonal hexagons within each underlying triangle. This design was used during the Seljuk Sultanate of Rum at the Çifte Minare madrasa in Erzurum, Turkey (late thirteenth century). Figure 107b also places octagons at the same locations, but their size is determined by bisecting the triangular edges at 1/3 intervals. This design also incorporates hexagons whose size is determined by their midpoints being placed upon the vertices of the generative triangles. This is likewise a Seljuk Sultanate of Rum pattern, and is found at the Cincikh mosque in Aksaray, Turkey (1220-30). The size of the hexagonal pattern elements in Fig. 107c is determined by their corresponding to the size of a regular hexagon placed within each underlying triangle [as per Fig. 105a]. However, this small hexagon is removed, and only the extended lines are kept [as per Fig. 105d]. The size of the superimposed nonagons is determined by their working together with the hexagons to create regular squares within each underlying square. This Seljuk sultanate of Rum design is from the Sungur Bey mosque in Nigde, Turkey (1335). Figure 107d places dodecagons at the centers of the generative hexagons: their size being determined by placing the midpoints of each dodecagonal edge upon the vertices of the generative tessellation. As stated, very similar designs comprised of superimposed dodecagons, but with slightly different proportions, can be created from different underlying polygonal structures [Figs. 97c and 99b]. The polygonal derivation in Fig. 107d is particularly successful in creating background elements that are more balanced in size and shape with one another, and conforms with a Tughluqid example from the Shah Rukn-i ‘Alam in Multan, Pakistan (1320-24) [Photograph 69].

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Fig. 104

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Fig. 105

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Fig. 106

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Fig. 107

Figure 108 illustrates six historical designs that are created from the 3.122 semi-regular tessellation of regular triangles and dodecagons. Figure 108a is a median pattern that employs 60° crossing pattern lines placed at the midpoints of the underlying polygonal edges. This is a very common threefold pattern, and early Seljuk examples include the east tower at Kharraqan (1067) [Photograph 17], and the Friday Mosque of Golpayegan, Iran (1105-18). An early Fatimid example is found at the Sayyid Ruqayya Mashhad in Cairo (1133). Multiple examples from the Seljuk Sultanate of Rum include the Great Mosque at Kayseri, Turkey (1205), and the Great Mosque at Akşehir near Konya (1213). A contemporaneous Ayyubid example is found at the Imam al-Shafi’i mausoleum in Cairo (1211). Multiple Mamluk examples include a window grille at the Ibn Tulun mosque in Cairo (1296), a door at the Vizier al-Salih Tala’i mosque in Cairo (1303), and the Amir Salar and Amir Sanjar al-Jawli complex in Cairo (1303-04). An Ilkhanid example from the same approximate period was used at the mausoleum of Uljaytu in Sultaniya (1307-13). Figure 108b shows a median pattern (by author) that uses 90° crossing pattern lines to create a pattern matrix comprised of superimposed dodecagons. While not known to the historical record, this design is appealing and certainly conforms to the aesthetics of this ornamental tradition. Figure 108c shows a curvilinear variation that comes from the 30-volume Quran (1313) commissioned by Sultan Uljaytu.Footnote 5 Figure 108d shows an obtuse pattern that includes the typical rosette treatment to the 12-pointed star inside each dodecagon [as per Fig. 222]. One of the earliest examples of this well-known design is a Seljuk raised brick panel from the southern iwan of the Friday Mosque at Forumad in northeastern Iran (twelfth century). An Ilkhanid example is found at the mausoleum of Uljaytu in Sultaniya, Iran (1307-13); and locations of later Mamluk examples include the Amir Aq Sunqar funerary complex in Cairo (1346-47) [Photograph 45], and the Sultan Qansuh al-Ghuri complex in Cairo (1503-05). This design can also be made from an underlying tessellation of dodecagons separated by barrel hexagons and trapezoids [Fig. 321j]. Figure 108e utilizes a Maghrebi variation to the added 12-fold rosette. This was used by Alawid artists at the Moulay Ismail Palace in Meknès, Morocco (seventeenth century). Figure 108f shows a two-point pattern with the same variety of added rosette to the 12-pointed star as in Fig. 108d. Examples of this design include an illuminated page from a Quran produced in Baghdad (1303-07) that was calligraphed by Ahmad ibn al-Suhrawardi and illuminated by Muhammad ibn Aybak ibn ‘Abdullah, and a Mamluk stone mosaic panel from the Amir Aq Sunqar funerary complex in Cairo (1346-47).

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Fig. 108

Figure 109 illustrates six designs created from the 4.6.12 semi-regular tessellation of squares, hexagons and dodecagons. Figure 109a shows a median pattern that employs 60° crossing pattern lines placed at the midpoints of the underlying polygonal edges. This is a Seljuk Sultanate of Rum pattern from the Hasbey Darül Huffazi madrasa in Konya (1421). Figure 109b shows an obtuse pattern derived from octagons placed within each square module, thus producing 135° crossing pattern lines at each midpoint of the underlying polygonal edge. This design is greatly enhanced by the underlying hexagons incorporating six-pointed star rosettes. This design can also be created from the dual of this grid (dashed lines), whereby the 5-, 6-, and 12-pointed stars are derived directly from the alternative underlying tessellation. This pattern enjoyed popularity among Mamluk artists, and examples include the Amir Sanqur al-Sa’di funerary complex in Cairo (1315); the Amir Ulmas al-Nasiri mosque and mausoleum in Cairo (1329-30); the Sultan Qansuh al-Ghuri madrasa (1501-03); and the Sultan Qansuh al-Ghuri Sabil Kuttab in Cairo (1503-04). Figure 109c shows an obtuse pattern with 120° crossing pattern lines at the underlying polygonal midpoints. This design approximates a Mamluk window grille at the Sultan Qala’un funerary complex in Cairo (1284-85). This same design can be created from an alternative underlying tessellation comprised of just triangles, squares and hexagons [Fig. 114b], with only slight differences in the proportions of the applied pattern lines. Figure 109d is unusual in that it employs the vertices of the underlying tessellation as determining coordinates for the pattern. This example is composed of just two sizes of superimposed hexagon, and is the product of artists working during the Seljuk Sultanate of Rum at the Izzeddin Keykavus hospital and mausoleum in Sivas (1217-18). Figure 109e shows a two-point pattern with superimposed octagons and dodecagons within the pattern matrix. The six-pointed stars at the centers of the underlying dodecagons are an arbitrary inclusion that is a sixfold corollary to the more common infill of ten-pointed stars [Fig. 224b]. This is from the Zuzan madrasa in northeastern Iran (1219); one of the few Khwarizmshah buildings still standing. Figure 109f shows a two-point pattern with the same dodecagonal motif as Fig. 109e, but with a very simple structure of parallel pattern lines that express the triangular repeat unit as well as its hexagonal dual. This is a Ghurid design from the minaret of Jam (1174-75 or 1194-95).

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Fig. 109

Figure 110 demonstrates a pattern created from the isometric 32.4.3.4-3.4.6.4 two-uniform tessellation of regular triangles, squares, and hexagons. This interweaving design is created simply by offsetting, and thereby widening the lines of the generative tessellation itself. As a consequence, the shapes of the background pieces conform to the polygonal modules of the underlying tessellation. This parallel offset process can be regarded as a variety of the two-point design derivation that utilizes perpendicular parallel lines placed within the square modules. This atypical Seljuk design is found at the western tomb tower at Kharraqan (1093). The two designs in Fig. 111 represent a standard two-point pattern and variation created from the isometric 32.4.3.4-3.4.6.4 two-uniform tessellation. These designs also employ the two sets of perpendicular parallel pattern lines within each underlying square module. The standard design in Fig. 111a extends the pattern lines from the underlying squares until they meet with other extended lines within the adjacent triangles and hexagons. Figure 111b replaces the six underlying triangles, six squares, and central hexagon that are located at the center of the panel with a single dodecagon. This variation produces the conditions for the central 12-pointed star. Both of these designs are found at the mausoleum of Muhammad Basharo in the village of Mazar-i Sharif in TajikistanFootnote 6 (1342-43).

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Fig. 110

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Fig. 111

Figure 112 represents two historical two-point patterns that have the distinctive fourfold rotational motif created from an underlying square contiguously surrounded by four underlying triangles. The underlying generative tessellation in Fig. 122a is a 32.4.3.4-3.4.6.4 two-uniform tessellation of triangles, squares, and hexagons. The resulting design in Fig. 112b is from the Khwarizmshahid madrasa in Zuzan, northeastern Iran (1219) [Photograph 38]. It is worth noting that as a variation, the central 12-pointed star shown in Fig. 111b can also be applied to the implied central dodecagon within the underlying hexagon and surrounding triangles and squares of this underlying tessellation. The underlying generative tessellation in Fig. 112c is a 33.42-32.4.3.4 two-uniform tessellation of just triangles and squares. The design in Fig. 112d is from the anonymous Persian language treatise On Similar and Complementary Interlocking Figures in the Bibliothèque Nationale de France in Paris, but is not known within the architectural record. Both of these designs have the same derivation of perpendicular pattern line applications within the square module as the isometric design in Fig. 111a.

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Fig. 112

Perhaps the most frequently used orthogonal arrangement of polygons for creating patterns from the system of regular polygons is the orthogonal 3.4.3.12-3.122 two-uniform tessellation of triangles, squares and dodecagons. Figure 113 illustrates six historical designs created from this underlying tessellation. Figure 113a shows a median pattern with 60° crossing pattern lines that was widely used throughout Muslim cultures. An early Ayyubid example is found at the Imam al-Shafi’i mausoleum in Cairo (1211), and a later Qara Qoyunlu example was used at the Great Mosque at Van (1389-1400). Figure 113b shows a Timurid variation of this median pattern from the Abu’l Qasim shrine in Herat, Afghanistan (1492), that employs an arbitrary treatment to the underlying square region. Figure 113c shows an obtuse pattern with 120° crossing pattern lines. The proportions of this example are consistent with the placement of regular hexagons within each underlying triangle. Multiple examples of this design exist within the architectural record. An early Zangid example was used on the wooden minbar from Aleppo commissioned by Nur al-Din in 1186.Footnote 7 Examples from the Seljuk Sultanate of Rum are found at the Sultan Han in Kayseri, Turkey (1232-36), and the Sultan Mesud tomb in Amasya, Turkey (fourteenth century). The Mamluks were particularly disposed toward this design and the many locations include: the Sultan al-Nasir Muhammad ibn Qala’un in the Cairo citadel (1295-1303); the Amir Sanqur al-Sa’di funerary complex in Cairo (1315); the Amir Altinbugha al-Maridani mosque in Cairo (1337-39); the Araq al-Silahdar mausoleum in Damascus (1349-50); an illuminated frontispiece for a Quran produced by Ya’qub ibn Khalil al-Hanafi in 1356; and an outstanding inlaid polychrome stone pavement in the Fort Qaytbey in Alexandria (1480s). Figure 113d shows a variation of this obtuse design with small arbitrary eight-pointed stars placed within the square modules. This Nasrid variation is from the Alhambra. Figure 113e shows a two-point pattern with 90° crossing pattern lines. This Mamluk design is found at the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81), as well as in the minbar of the Amir Azbak al-Yusufi complex in Cairo (1494-95) [Photograph 46]. The relative closeness in location and date invites the possibility that the same artist or atelier produced these two examples, or that the latter was copied from the former. Figure 113f is an unusual median pattern with an additive six-pointed star applied within alternating underlying dodecagons. This additive motif follows the fivefold convention that was occasionally applied to ten-pointed stars [Fig. 224b]. This Mamluk variation is from the stone minbar in the Zawiya wa-Sabil Faraj ibn Barquq in Cairo (1400-11).

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Fig. 113

Figure 114 illustrates three closely related patterns created from the 34.6-33.42-32.4.3.4 three-uniform tessellation of triangles, squares, and hexagons. Each of these is a median pattern with 60° crossing pattern lines. The grey polygons in the underlying generative tessellations are passive in that they do not actively contribute to the design process. Figure 114a can also be created from the 63 underlying tessellation [Fig. 98d]. This is an early design that was used by Fatimid artists on the side panels of two wooden minbars: that of the Haram al-Ibrahimi in Hebron, Palestine (1094), and that of the al-Amri mosque in Qus, Egypt (1156). Figure 114b is identical to Fig. 114a except that it adds superimposed large hexagons into the pattern matrix. The size and location of these are determined by 60° crossing pattern lines within the underlying triangles that are coincident with the underlying squares. This is an early Mamluk design from the Sultan Qala’un funerary complex in Cairo (1284-85). As mentioned above, a design with similar proportions can be produced from an alternative 4.6.12 tessellation [Fig. 109c]. However, the precise proportions of the design from the Sultan Qala’un funerary complex match the example created from the 34.6-33.42-32.4.3.4 three-uniform tessellation in this illustration. Figure 114c is identical to Fig. 114b except that the pattern lines within the passive squares have been trimmed to produce the irregular eight-pointed motifs with two perpendicular lines of reflected symmetry. This design can also be created from the 3.6.3.6 tessellation [Fig. 101b]. This design was used in the spandrel of the Fatimid wooden mihrab from the Sayyid Ruqayya Mashhad in Cairo (1133).Footnote 8

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Fig. 114

Figure 115 demonstrates a pattern created from the 36-33.42-32.4.12 three-uniform tessellation of triangles, squares and dodecagons. This is a median pattern with 60° crossing pattern lines. The grey triangles are passive, and the pattern lines contained within them are arbitrarily determined. The size of the large superimposed dodecagons within the pattern matrix is determined by their edges intersecting with the midpoints of the underlying square modules. This is a Qara Qoyunlu design from the Great Mosque at Van in Turkey (1389-1400).

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Fig. 115

Patterns created from the system of regular polygons occasionally employ underlying polygonal modules that are not regular. The most frequently used non-regular polygon is the distinctive ditrigonal shield module comprised of three 90° and three 150° included angles, with the angular proportions of overlapping squares in threefold rotation. Figure 116 illustrates ten examples of tessellations that employ this module. Figures 116a and b demonstrate how the ditrigon can be an interstice region within a tessellation of otherwise regular polygons. The other examples employ the ditrigon as a tessellating entity of equal formative merit to the regular polygons within the tessellation. These ten examples demonstrate the effectiveness of this module in creating tessellations with repetitive structures that are isometric, orthogonal, rhombic, rectangular, and elongated hexagons. However, only three of these ten tessellations appear to have been used historically: Figs. 116a, e and f. Figure 117 shows the isometric median design created from the tessellation represented in Fig. 116a. This design was used by artists working in multiple Muslim cultures, including the Seljuk Sultanate of Rum at the Yelmaniya mosque in Cemiskezck, Turkey (1274); the Mamluks at the Aqbughawiyya madrasa (1340) in the al-Azhar mosque in Cairo; and the Ottomans in their restoration of the Dome of the Rock in Jerusalem.

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Fig. 116

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Fig. 117

Figure 118 illustrates four historical patterns that can be produced from an isometric tessellation comprised of the ditrigonal modules in sixfold rotation around a six-pointed star interstice region. It is interesting to note that the underlying generative tessellation in the first three of these is itself a well-known geometric pattern created from the system of regular polygons [Fig. 95c]. Each of these four patterns is a variation on the same theme. Figure 118a shows the earliest such design, and possibly the prototype for later examples. This is from the Seljuk northeast dome chamber of the Friday Mosque at Isfahan (1088-89) [Photograph 19]. Figure 188b represents essentially the same pattern, but with slightly different placement of the pattern lines. This is a Mamluk design from a window grille at the al-Anzar mosque in Cairo. Patterns created from this underlying tessellation were also produced during the Seljuk Sultanate of Rum, and Fig. 118c shows a design from the portal of the Huand Hatun complex in Kayseri (1237), as well as the mihrab of the Ahi Serafettin mosque in Ankara (1289-90). The underlying tessellation of Fig. 118d further populates the interstice six-pointed stars with six pentagons that surround a central hexagon, thereby creating the conditions for the five- and six-pointed stars within these regions. This design is from the mihrab of the Karatay madrasa in Antalya, Turkey (1250).

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Fig. 118

Figure 119 shows the orthogonal median design produced from the tessellation in Fig. 116e that includes a fourfold rotation of the underlying ditrigons that fill alternating dodecagonal cells within the underlying generative tessellation. An unusual feature of the crossing pattern lines is the predominant use of 60° angular openings, and the introduction of 45° angular openings along the coincident edges of the four ditrigonal modules. This produces the visually pleasing regular octagons at the vertex where the four ditrigons meet. This Mamluk geometric pattern is from the side panels of the wooden minbar at the Vizier al-Salih Tala’i mosque in Cairo (c. 1300). The underlying generative tessellation in Fig. 120 is the third historical example from Fig. 116f, and is like that of Fig. 119 except that it places the four edge-to-edge ditrigons into each of the dodecagonal regions rather than every other one. Without the ditrigons, and associated four triangles, this would be the 3.4.3.12-3.122 two-uniform generative tessellation [Fig. 90]. The size of the large dodecagon within the pattern matrix is determined by their transecting the midpoints of the triangle and square modules of the generative tessellation. The earliest known use of this exceptional pattern was by Ghurid artists at both the minaret of Jam, Afghanistan (1174-75 or 1194-95), and the Shah-i Mashhad in Gargistan, Afghanistan (1176), and later examples include the work of Seljuk Sultanate of Rum artists at the Alaeddin mosque in Konya (1219-21).

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Fig. 119

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Fig. 120

The incorporation of underlying ditrigons always contributes a distinctive visual character to geometric patterns. Figure 121 illustrates six ahistorical designs (by author) from underlying generative tessellations that include this module. These examples are representative of a very large number of tessellations that can be produced with this added module; each of which will produce designs in all four of the pattern families. These six examples are all median patterns with 60° crossing pattern lines.

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Fig. 121

Generally, the ditrigonal shield is the only non-regular element occasionally incorporated into the system of regular polygons. An exception is found in a design produced during the Seljuk Sultanate of Rum, and speaks to the experimental approach to pattern making during this highly innovative period. Figure 122 illustrates a pattern created from an underlying tessellation of triangles and squares, as well as irregular pentagons that are clustered in fourfold rotation. The proportions of these pentagonal elements are determined by simply dividing the interstice regions created from the orthogonal distribution of the triangles and squares into four pieces. Unlike the ditrigonal modules, these pentagons do not tessellate in other configuration, and as such are nonsystematic. This unusual example is an orthogonal median design located at the tomb of Seyit Mahmut Hayrani in Aksehir near Konya (1275).

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Fig. 122

3.1.2 Octagon and Square Patterns

As mentioned previously, while the octagon is a regular polygon and would appear to qualify as a member of the set of modules that comprise the system of regular polygons, the fact that this will only tessellate with other regular polygons in the single 4.82 arrangement makes this a special case that, for the purposes of greater clarity, is treated herein as a separate design category. While the singular 4.82 arrangement of underlying polygons is, on the one hand, limiting to the tessellating process, this arrangement of octagons and squares is responsible for more individual designs than any other single polygonal tessellation. Figure 123 demonstrates these two polygonal modules with a variety of applied pattern lines that were used to a greater or lesser extend by artists throughout Muslim cultures. As shown, multiple variations of pattern line application are possible within each of the four pattern families, and the specific examples shown are by no means exhaustive. What is more, some designs do not place pattern lines within the square modules, and others will employ alternating octagonal edges for pattern line placement.

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Fig. 123

Figure 124 illustrates the basic design from each of the four pattern families. Figure 124a shows the well-known acute pattern with 45° crossing pattern lines. Figure 124b shows the classic star-and-cross median pattern with 90° crossing pattern lines. This is one of the earliest and most widely employed Islamic geometric patterns, and is easily created from the point-to-point orthogonal arrangement of the eight-pointed stars. Figure 124c shows the obtuse pattern with 135° crossing pattern lines. This was also used with great frequency throughout Muslim cultures, and can similarly be produced very easily through a corner-to-corner orthogonal arrangement of the octagons. Figure 124d shows a less well-known two-point pattern with 45°/135° supplementary angles of the pattern lines placed at two points upon each polygonal edge. A relatively early example of the acute pattern in Fig. 124a is from a panel of a metal door at the Sultan Qala’un funerary complex in Cairo (1284-85). The number of classic star-and-cross median designs in Fig. 124b is too numerous to elucidate herein, but some of the earliest examples include a panel from the Abbasid minbar at the Great Mosque of Kairouan (c. 856); one of the Tulunid arch soffits at the ibn Tulun mosque in Cairo (876-79); a Yu’firid ceiling panel from the Great Mosque of Shibam Aqyan near Kawkaban, Yemen (pre-871-72); a carved stucco panel from the No Gumbad mosque in Balkh, Afghanistan (800-50) [Photograph 10]; a Buyid carved stucco border that surrounds the mihrab at the Friday Mosque at Na’in, Iran (960); and a Saminid carved stucco panel from the Sabz Pushan outside Nishapur (960-85). The obtuse pattern in Fig. 124c has been found in the pre-Islamic Great Temple of Palmyra (c. 36), and a particularly pleasing example produced during the Seljuk Sultanate of Rum is found at the Esrefoglu Süleyman Bey in Beysehir, Turkey (1296-97). The two-point example in Fig. 124d is less common, and a fine example is from a Nasrid silk brocade textile at Metropolitan Museum of Art in New YorkFootnote 9 (fourteenth century).

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Fig. 124

Figure 125 illustrates four historical variations to the acute pattern created from the underlying 4.82 tessellation. Figure 125a removes all pattern lines from the underlying square module. This also opens up the eight-pointed stars through a simple subtractive process. This is a Seljuk variation from the minaret of Hotem Dede in Malatya, Iran (twelfth century). Figure 125b makes eight-pointed stars out of the otherwise four-pointed stars within the underlying square modules. This is an Ayyubid variation from the Sahiba madrasa in Damascus (1233-45). Figure 125c also places eight-pointed stars within the underlying square modules, but extends the lines of the eight-pointed stars until they meet with other extended pattern lines, thus creating a very distinctive design with two varieties of eight-pointed star. This is a Seljuk design from the Gunbad-i Alayvian in Hamadan, Iran (late twelfth century) [Photograph 22]. Figure 125d incorporates a swastika motif into the parallel lines of Fig. 125c. The overall aesthetic results from an interlocking treatment emphasized through color contrast. This is a Seljuk Sultanate of Rum design variation from the Sirçali madrasa in Konya (1242-45).

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Fig. 125

Figure 126 illustrates eight variations to the median design of Fig. 124b. Figure 126a employs 60° crossing pattern lines rather than the more typical 90° of the classic star-and cross pattern. This is an Ayyubid variation from the Firdaws madrasa in Aleppo (1235-36). Figure 126b places eight-pointed stars within the square modules, and extends the pattern lines within the octagonal modules from midpoint to midpoint of the underlying polygonal edges, thereby transforming the more common eight-pointed star into an octagon. This is a Nasrid variation from a wooden ceiling at the Alhambra. Figures 126c and d place a ring of squares within the octagonal modules. Figure 126c is an Umayyad variation from the mosaics at the Great Mosque of Córdoba (971), and Fig. 126b is a Timurid variation from the Ghiyathiyya madrasa in Khargird, Iran (1438-44). Figure 126e arbitrarily adds eight-pointed stars within the square modules and an eight-pointed star rosette within the octagons. This is a Marinid variation from the Bu ‘Inaniyya madrasa in Fez (1350-55). Figure 126f is a frequently used subtractive pattern that radically disguises its star-and-cross origins, and historical examples include A Qarakhanid variation is found at the Maghak-i Attari mosque in Bukhara (1178-79); the Zangid minbar at the al-Aqsa mosque (1187); and a Mamluk door (1303) at the Vizier al-Salih Tala’i mosque in Cairo. The design variations in Figs. 126g and h are typical to the Maghreb. Examples of both these variations are found at the Bu ‘Inaniyya madrasa in Fez (1350-55).

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Fig. 126

The six designs in Fig. 127 are historical variations to the standard obtuse pattern in Fig. 124c. Figure 127a extends the lines of the standard obtuse design into the underlying square module, creating smaller octagons within the pattern matrix. This is an Ilkhanid variation from a Quranic Frontispiece dated 1304. Figure 127b shows a simple Buyid curvilinear variation from the Friday Mosque at Na’in, Iran (960). Figures 127c and d use alternating edges of the underlying octagons for pattern line placement. Figure 127c can also be created as a two-point pattern [Fig. 128d], and this design was used by Ibn al-Bawwab in his celebrated Baghdad Quran produced in 1001 [Photograph 6]. Figure 127d is similar to that of Fig. 127c in its use of alternating midpoints of the underlying octagons. This appears to have first been used by Fatimid artist in a wooden mihrab from the mausoleum of Sayyidah Nafisah in Cairo (1138-46), and later by Mengujekid artists at the Great Mosque of Divrigi, Turkey (1228-29). A Mamluk variation was used on the minaret at the Sultan Qaytbay funerary complex in Cairo (1472-74). Figure 127e employs bilateral concave octagonal motifs that are common to the fourfold system A. This is a Seljuk variation from an unattributed stucco panel at the Tehran Museum. Figure 127f places arbitrary eight-pointed stars within each underlying square module, and reflects four of the octagonal angles within the underlying octagons to produce large four-pointed stars. This variation comes from the Mughal pavement at the Taj Mahal in India (1632-53).

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Fig. 127

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Fig. 128

Figure 128 illustrates eight historical two-point patterns created from the 4.82 tessellation of squares and octagons. Figure 128a uses alternating underlying polygonal edges for pattern line placement, with 45°/135° supplementary angles. This is a Maghrebi design that appears to have first been used by the Almohads at the al-Kutubiyya mosque in Marrakech, Morocco (twelfth century), and later in the Nasrid carved stucco ornament of the Alhambra. Figure 128b also uses alternating edges of the underlying octagon for pattern line placement, and this was also used at the Alhambra. Figures 128c and d place two sets of 45° crossing pattern lines at each of the two points of alternating underlying octagonal edges. The earliest known use of the pattern in Fig. 128c is from a Ghurid carved stucco panels from Lashkar-i BazarFootnote 10 in Afghanistan (after 1149), and a later Seljuk Sultanate of Rum example is from the Karatay madrasa (1251-55). Figure 128d shows an alternative method for creating the design used by Ibn al-Bawwab as per Fig. 127c above [Photograph 6]. Figures 128e and f also employ alternating edges, and differ only in the treatment of the underlying square module. Both variations are Mengujekid: Fig. 128e being from the Divrigi hospital in Turkey (1228-29), and Fig. 128f from the nearby Great Mosque of Divrigi (1228-29). Figure 128g employs pattern lines that are perpendicular to the underlying polygonal edges. This design is also from the Great Mosque of Divrigi, and it is possible that these three examples were created by the same artist. Figure 128h is derived from an assortment of pattern line features that all transect the two points of each underlying polygonal edge. These include squares, double sets of 60° crossing pattern lines, and large dodecagons. The success of this pattern is the result of offsetting the pattern lines in a single direction rather than the more common practice of widening lines equally in both directions. This exceptionally successful Ghurid design is from the raised brick ornament of the Friday Mosque at Herat (1200).

Figure 129 represents four historical design variations that orient their pattern lines upon the vertices of the 4.82 tessellation rather than points on the underlying polygonal edges. While this alternative design practice is atypical, it will occasionally produce patterns that are very successful. The design in Fig. 129a is comprised of two sizes of square as well as eight-pointed stars. The earliest occurrence of this pattern is on one of the Tulunid carved stucco arch soffits at the ibn Tulun mosque in Cairo (876-79) [Photograph 9]. Later locations include the Almohad al-Kutubiyya mosque in Marrakech, Morocco (twelfth century), and the Nasrid Alhambra. Figure 129b is very similar except for the discontinuous lines that break with standard methodological practices. However, it is nonetheless handsome in its overall composition. This is an Artuqid pattern from the Great Mosque of Dunaysir in Kiziltepe, Turkey (1204). Figures 129c and d achieve their interweave by widening the pattern lines in a single direction. These designs can also be created from the two-point process [Fig. 128g], in which case the pattern lines are widened in both directions until the three lines meet at a single point. A variation of the design in Fig. 129c (with scallops incorporated into the pattern matrix) was used in a window grille by the Umayyads in Córdoba (980-90). Later Ghurid examples include the raised brick ornaments of the minaret of Jam in Afghanistan (1174-75 or 1194-95), and the western mausoleum at Chisht, Afghanistan (1167). The very similar design in Fig. 129d was used in the Mudéjar ornament at the Alcazar in Seville (1364-66), and by the Nasrids at the Alhambra.

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Fig. 129

3.1.3 Fourfold System A

The fourfold system A is comprised of a limited number of polygonal modules that can tessellate in an unlimited number of combinations. Figure 130 illustrates the polygonal modules of this system, along with their associated pattern lines in each of the four pattern families. It is important to note that the nine polygonal modules represented in this figure are not exhaustive, and that additional modules are occasionally employed within this system. These secondary modules are often derived via interstice regions through the process of tessellating with the otherwise standard modules. The applied pattern lines in this system, as well as the other historical systems that follow, are generally more formalized than that of the system of regular polygons. This is due in part to the fact that the modules of the system of regular polygons can tessellate both isometrically and orthogonally, eliciting a broad range of associated angular openings for the crossing pattern lines of the various pattern families: e.g., 30°, 45°, 60°, 90°, 120°, and 135° [Fig. 92]. By contrast, the standardized angular openings of the fourfold system A employ fewer angles: 45°, 90° and 135°. Another reason for the greater diversity of pattern line application to the system of regular polygons is the earlier provenance of this system, with earlier examples being produced during a period of greater exploratory experimentation and prior to the generalized standardization that came with the maturity of this ornamental tradition.

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Fig. 130

There are three edge lengths among the polygons of the fourfold system A. This forces the polygonal modules to associate with one another based upon edge determinants. It is worth noting that the only regular polygons in the fourfold system A are the square and two sizes of octagon. The applied pattern lines of the larger octagon and square modules are identical to four examples from the previous section detailing patterns created from the 4.82 tessellation. Indeed, such patterns can be equally regarded as part of the system of regular polygons or the fourfold system-A. This overlap is all the more reason for this 4.82 variety of design to be given its own categorization.

Figure 131 demonstrates methods for constructing the polygonal modules of the fourfold system A using the large octagon as the foundation from which each additional module is derived. Figure 132 provides the proportional relationships of the three edge lengths as they relate to the foundational octagon. The image on the right represents the edge lengths as nested squares. The indicated proportional relationship of 1:1.4142… between the short and long edges is √2.

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Fig. 131

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Fig. 132

Figure 133 illustrates the seven rotational combinations of the triangle from the fourfold system A, with applied pattern lines from the acute family. The outer long edges of the rotated clusters of triangles must be coincident with the long edges, be they other triangles in 180° rotation (as per the cluster of four triangles), the long edges of the trapezoids, or a square interstice module (as per the cluster of six triangles). The demonstrated arrangements of clustered triangles in 45° rotational increments, with coincident triangles, trapezoids, or square, create pattern motifs (in this case associated with the acute family) that are well known to this ornamental tradition.

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Fig. 133

Figure 134 demonstrates the dualing characteristic between the underlying tessellations created from the fourfold system A. A remarkably feature of this system provides for each tessellation to have a dual relationship with another tessellation comprised of polygonal modules from this same system. As a consequence, it is possible to create a specific pattern from either of the two dualing underlying tessellations. In this example, the two tessellations are identical in every respect except that their locations shift between the rectangular repeat unit (solid black line) and the rectangular dual of the repeat unit (dashed black line). In examples such as this, the dual tessellations are reciprocals. Figure 135 illustrates three patterns (by author) from the acute, median and obtuse families that are created from the dualing underlying reciprocal tessellations from Fig. 134. Although made from dual tessellations, the obtuse pattern in Fig. 135a and the acute patterns in Fig. 135b are identical: the obtuse pattern being created from 135° angular openings (where the bisector of the angle is perpendicular to the polygonal edge), and the acute pattern being produced from 45° angular openings. Similarly, the acute pattern in Fig. 135e and the obtuse pattern in Fig. 135f are also identical. What is more, the acute pattern in Fig. 135b is the reciprocal of the acute pattern in Fig. 135e; and the obtuse pattern in Fig. 135a is the reciprocal of the obtuse pattern in Fig. 135f. Acute and obtuse patterns created from the fourfold system A are demonstrably correlated, and without knowing the specific underlying tessellation of a given historical example, patterns produced from 45° and 135° angular openings can equally be regarded as either acute or obtuse. By contrast, the 90° crossing pattern lines of the median patterns in Figs. 135c and d produce the same design with the same relative location in both dualing tessellations.

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Fig. 134

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Fig. 135

Designs created from tessellations made up of just a single underlying polygonal module are always highly repetitive, and lacking primary star forms. These invariably qualify as field patterns. Figure 136 illustrates four such patterns created from just the small hexagon of the fourfold system A. Figures 136a and c demonstrate the reciprocal relationship between acute and obtuse patterns common to this system; and in this case the orientation of the pattern is rotated 90°. Figure 136b shows a well-known median pattern, with an early Anatolian Seljuk example used at the Great Mosque of Niksar in Turkey (1145). The two-point pattern in Fig. 136d is by the author, and not known to the historical record. The six field patterns in Fig. 137 are created from the underlying tessellations comprised of just the large hexagon from this system. Figure 137a shows an acute design (by author) with 45° crossing pattern lines, while the variation in Fig. 137b (by author) incorporates two-point lines upon the horizontal polygonal edges into the otherwise acute design. Figure 137c shows a conventional median design (by author) with 90° crossing pattern lines, while Fig. 137d shows a historical median pattern that employs 60° crossing patterns lines. The use of 60° angular openings within the fourfold system A is unusual. This is a Seljuk example from the Kwaja Atabek mausoleum in Kerman, Iran (1100-1150). Figure 137e shows an obtuse pattern (by author), and Fig. 137f shows a two-point pattern (by author).

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Fig. 136

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Fig. 137

The majority of field patterns associated with the fourfold system A were produced from underlying tessellations comprised of two or more polygonal modules. Figure 138 shows six field patterns created from just two modules: the square and large hexagon. Figure 138a shows a Seljuk acute pattern from the mihrab of the Ibrahim mosque at Salihin in Aleppo (1112). This was also used by the Saltukids on the Tepsi minaret in Ezurum, Turkey (1124-32). The acute variation in Fig. 138b is a Ghaznavid design from the Ribat-i Mahi near Mashhad, Iran (1019-20) [Photograph 14]. This arbitrarily changes the pattern lines associated with the underlying square module. Figure 138c shows a very-well-known median design that was used in one of its earliest locations at the eastern tomb tower at Kharraqan (1067-68). The median variation in Fig. 138d is a Qara Qoyunlu design from the Great Mosque of Van (1389-1400). Figure 138e shows an obtuse design (by author) that is a matrix of superimposed octagons. And the obtuse variation in Fig. 138f is a Qarakhanid design from the Maghak-i Attari mosque in Bukhara, Uzbekistan (1178-79). This design is entirely comprised of superimposed dodecagons. Figure 139 shows a further variation of the median pattern from Fig. 138c. This design has both varieties of applied pattern from Figs. 138c and d within a single construction. This was produced during the Seljuk Sultanate of Rum and is from the mihrab of the Great Mosque of Erzurum (1179). The underlying polygonal tessellation in Fig. 140 is comprised of large hexagons and pentagons, and the repeat unit for this median pattern is either of the two hexagons illustrated. These are duals of one another. The resulting pattern is a network of superimposed non-regular octagons. This was produced during the Seljuk Sultanate of Rum and is found at multiple locations in Turkey, including the Sultan Han in Kayseri (1232-36), and the Esrefoglu Süleyman Bey in Beysehir, Turkey (1296-97). The field pattern in Fig. 141 employs four generative polygonal modules: the large hexagons, the small hexagon, the pentagon, and a small rhombic interstice element. The earliest example of this median design is from the Ildegizid mausoleum of Yusuf ibn Kathir in Nakhichevan (1161-62), and two other examples were used during the Seljuk Sultanate of Rum: at the Haunt Hatun in Kayseri (1238), and the Haci Kiliç mosque and madrasa in Kayseri (1249). Figure 142 demonstrates how local regions of an otherwise repetitive field pattern can be changed to include eight-pointed stars. Selected modules from the underlying tessellation in Fig. 139 have been replaced with pentagons and the large octagon so that eight-pointed stars are introduced into the pattern matrix. This example is also the product of artists working in the Seljuk Sultanate of Rum, and is from a wooden balustrade at the Esrefoglu Süleyman Bey in Beysehir, Turkey (1296-97).

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Fig. 138

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Fig. 139

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Fig. 140

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Fig. 141

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Fig. 142

There are a large number of patterns created from the fourfold system A that have as their basis an orthogonal arrangement of underlying large hexagons. Figure 143 demonstrates how this layout of the large hexagons produces eight-pointed star interstice regions. In point of fact, this underlying tessellation is essentially the same as the classic star and cross design produced from the 4.82 generative tessellation [Fig. 124b]. This eight-pointed star interstice region produces very acceptable pattern elements in each of the four pattern families. The two examples in this illustration were both produced during the Seljuk Sultanate of Rum. Figure 143a represents a median design from the Great Mosque of Erzurum, Turkey (1179). Ghurid artists employed this same design just 18 years later at the eastern mausoleum at Chisht, Afghanistan (1197). The variation in Fig. 143b is from the Gök madrasa and mosque in Amasya, Turkey (1266-67). The patterns in Fig. 144 employ the same underlying tessellation. Figure 144a shows a median pattern with atypical 60° crossing pattern lines. This design was used during the Muzaffarid period at the Friday Mosque at Kerman (1349). This design is similar to a Chaghatayid or Sufid example from the Task-Kala caravanserai in Konye-Urgench, Turkmenistan (fourteenth century) [Fig. 147b]. Figure 144b shows an obtuse pattern (by author) with an additive eightfold rosette within each eight-pointed star. This design is geometrically similar to a Qarakhanid design from the Maghak-i Attari mosque in Bukhara, Uzbekistan (1178-79) [Photograph 16], in that both include superimposed octagons set upon the vertices and centers of their respective square repeats. However, the octagons in Fig. 144b are all of the same size, whereas those in the Qarakhanid example are of two different sizes [Fig. 151]. Figure 145 illustrates two underlying generative tessellations for the same median pattern. This is a very-well-known design employed widely throughout Muslim cultures. The underlying tessellation on the right side of this figure fills the large eight-pointed star interstice regions of the previous examples with underlying large octagons surrounded by eight pentagons as per Fig. 142. The earliest occurrences of this design are from the raised brick ornaments of the Seljuks and Ghurids in Khurasan, including the minaret of the Friday Mosque at Damghan, Iran (1080); an example that is incorporated into a Kufi inscription at the Friday Mosque at Golpayegan, Iran (1105-18); the minaret of Daulatabad in Afghanistan (1108-09) [Photograph 20]; the minaret of the Friday Mosque at Saveh, Iran (1110); the Friday Mosque at Sangan-e Pa’in, Iran (late twelfth century); and the minaret of Jam in central Afghanistan (1174-75 or 1194-95). This is also illustrated in the anonymous treatise On Similar and Complementary Interlocking Figures. Footnote 11 Later examples of this ever-popular design include the Shaybanid polychromatic brick ornament at the Shir Dar madrasa in Registan Square, Samarkand (1619-36) [Photograph 72]; the Mughal stone mosaics of the mausoleum of Akbar in Sikandra, India (1613); and the Ottoman ornament of the Bayt Ghazalah private residence in Aleppo (seventeenth century). The underlying tessellation in Fig. 146 also incorporates large octagons surrounded by eight pentagons. In this case the ring of pentagons is separated by small hexagons located at the midpoints of the repetitive edges. This produces a non-regular interstice dodecagon at the centers of the square repeat into which the design’s cruciform element is located. The angular openings of the crossing pattern lines in Fig. 146 are of two types: 45° and 67.5°. This is a Seljuk Sultanate of Rum design from the Çifte madrasa in Kayseri (1205).

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Fig. 143

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Fig. 144

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Fig. 145

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Fig. 146

The placement of eight underlying squares in rotation is a device that was used with some frequency for creating designs from the fourfold system A. The pattern lines in Fig. 147a are almost identical to those of Fig. 146, but produced from an underlying tessellation that places eight squares and eight rhombi in eightfold rotation at the vertices of the square repeat unit. Indeed, both these two designs can be produced from either underlying tessellation. The design in Fig. 147a is unusual in that it employs three varieties of crossing pattern line placed at the midpoints of the underlying polygonal edges: 45°, 67.5°, and 90°. This is a Seljuk pattern that surrounds the portal of the Friday Mosque at Gonabad (1212). The design in Fig. 147b is also unusual. The underlying tessellation does not make use of the eight rhombi in rotation, and these areas now become eight-pointed star interstice regions. The application of the pattern lines within these regions of the underlying tessellation is unconventional. The entire pattern matrix is determined by the placement of eight regular hexagons around the interstice eight-pointed stars that are centered on the vertices of the square repeat unit (dashed lines). This results in the hexagons being located upon the vertices of the 4.82 tessellation of octagons and squares (white lines). These hexagons are aligned to the indicated sets of eight radii at every other underlying interstice eight-pointed star. While the proportion of the resulting eight-pointed stars within these regions is not ideal, the extended lines of the regular hexagons serendipitously produce regular octagons within the alternating underlying eight-pointed stars at the centers of the repeat units. The size and placement of each hexagon are determined by placing two of the vertices of each hexagon at the midpoints of the underlying squares. Extending the lines of the hexagons to intersection points of other extended lines completes this highly unusual design. This is either a Sufid or Chaghatayid pattern that is found on the arch soffit of the entry iwan at the Task-Kala caravanserai in Konye-Urgench, Turkmenistan (fourteenth c). The median pattern in Fig. 148 has shared characteristics with the median design in Fig. 139, but with eight-pointed stars and nested octagons at the nodal centers. This design fills the large eight-pointed star interstice region in Figs. 143 and 144 with the eight squares and rhombi in rotation as per the previous examples, and the resulting median pattern is, in fact, a variation of the design in Fig. 143a, the only difference being the added small octagons that result from the ring of eight underlying squares. This Seljuk Sultanate of Rum design is located at the Alay Han near Aksaray (1155-97). Figure 149 demonstrates the construction of a pattern that utilizes the ring of eight squares that adhere to the Maghrebi stylistic conventions. Figure 149a employs the standard applied pattern lines associated with the median family to each of the underlying polygonal modules except for the squares. These applied pattern lines place notches in the standard applied squares so that they become crosses. This design (by author) is an adequate example of a median pattern in the western style produced from the fourfold system A. However, Fig. 149b demonstrates how further arbitrary modifications to the cruciform pattern lines in the underlying square modules will create a more interesting variation. This example is Nasrid and was used in the Palace of Myrtles at the Alhambra (1370). Both of these treatments to the applied pattern lines associated with the underlying squares were occasional features of the Maghrebi use of this design system. Figure 150 illustrates two patterns created from underlying tessellations wherein the rings of eight squares are rotated 22.5° from the previous examples. This produces a tessellation of squares, rhombi, and large octagons. Each of the two historical examples created from this tessellation are provided with additive swastika motifs incorporated into the pattern matrix. Figure 150a can also be created through modifying the acute pattern produced from the underlying tessellation of squares and octagons [Fig. 125d]. This example was produced during Seljuk Sultanate of Rum and is from the Karatay madrasa in Konya (1251-55). The swastika design in Fig. 150b is distinctive in that it incorporates the generative tessellation itself as the basis for the completed design, and is derived by applying the swastika motifs into each of the square modules of the generative tessellation. This is a Timurid design from a marble column at the Gawhar Shad madrasa and mausoleum in Herat, Afghanistan (1417-38).

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Fig. 147

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Fig. 148

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Fig. 149

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Fig. 150

The design in Fig. 151 incorporates two varieties of crossing pattern line: 45° and 90°. These angles are determined by the pattern line distribution within the underlying triangular modules. These two types of crossing pattern line produce both the small four-pointed stars that are typical to acute patterns within this system, and the large eight-pointed stars that are typical to median patterns. By being equal to the short edge of the triangles, the edge length of the underlying squares does not conform to the typical size of the square modules from this system. However, this smaller square module works well in the context of the two varieties of crossing pattern line. Similarly, the length of the long edges of the triangular module determines the size of the underlying octagon, and this is also atypical to the proportions of the octagonal modules within the fourfold system A. This is a Qarakhanid pattern used on the façade of the Maghak-i Attari mosque in Bukhara, Uzbekistan (1178-79) [Photograph 16]. Figure 152 illustrates another pattern created from this unusual underlying tessellation. This is a two-point pattern produced during the Seljuk Sultanate of Rum that is from the Sirçali madrasa in Konya, Turkey (1242-45). The distinctive pattern line treatment within the underlying square element is an arbitrary inclusion.

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Fig. 151

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Fig. 152

Figure 153 shows an orthogonal design from the fourfold system A with standard median pattern line application to the underlying polygons. This design was used by the Seljuks in the Malik mosque in Kerman (eleventh to twelfth century), as well as the Mengujekids in the mausoleum of Behram Shah in Kemah, Turkey (1228).

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Fig. 153

Most patterns created from the fourfold system A repeat upon the orthogonal grid. While the example in Fig. 154 is no exception, this design utilizes the 4.82 arrangement of squares and octagons as its repetitive schema. Although relatively uncommon, the use of multiple repetitive elements—in this case squares and octagons—within a single design was also applied to the fourfold system B and the fivefold system. These forms of hybrid repeat provide further means of creating ever-greater pattern complexity and visual interest. The underlying generative octagons in this example are placed edge to edge at each vertex of the 4.82 grid. Octagons, pentagons, and squares provide further polygonal infill of the generative tessellation. Several examples of this design are known to the historical record, including a Seljuk Sultanate of Rum stone border at the Zeynebiye madrasa in Hani, Turkey (eleventh to twelfth centuries); the Mamluk minaret of the Attar mosque in Tripoli, Lebanon (1350); the border surrounding the early Ottoman mihrab in the Hatuniye madrasa in Karaman, Turkey (1382); and the Mughal tomb of I’timad al-Daula in Agra (1622-28) [Photograph 73]. Figure 155 illustrates another orthogonal median design created from this system that uses the 4.82 arrangement of squares and octagons as its repetitive schema. Except for the applied pattern lines into the underlying octagons that rest upon the vertices of the repetitive octagon, this design is much the same as that of Fig. 154. The applied pattern lines within these modules run parallel to the octagons of the underlying generative tessellation. This is a Qarakhanid design from the stone façade of the Maghak-i Attari mosque in Bukhara, Uzbekistan (1178-79). Figure 156 shows another median design with a hybrid repetitive structure. This example employs oscillating squares and rhombi in a highly eccentric repetitive structure. The included angles of the rhombi are 64.4712…° and 115.5288…°, and do not align with the inherent geometry of either the underlying tessellation or the pattern itself. Due to this, the eight-pointed stars that are located on the vertices of the square and rhombic grid are not in alignment with the grid itself. As with other oscillating square patterns, this design is orthogonal, but with a comparatively large amount of geometric information within each square repeat unit. This remarkable design is testament to the inventive skills of geometric artists working in the Seljuk Sultanate of Rum. This example comes from the mihrab of the Huand Hatun complex in Kayseri, Turkey (1237-38), and is also represented in the Topkapi Scroll.Footnote 12 The unusual character and repetitive complexity of this design suggest the possibility that the compiler of the Topkapi Scroll may have had either direct or indirect knowledge of the example in Kayseri.

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Fig. 154

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Fig. 155

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Fig. 156

The square is by far the most common repeat unit for patterns with fourfold symmetry. However rhombic repeat units that have 45° and 135° included angles were also occasionally employed. Figure 157 depicts three designs with this repetitive structure that are created from the fourfold system A. The underlying tessellation for these examples places four large octagons at the vertices of each rhombus. These are separated along the sides of each rhombus by a pentagon. Figure 157a represents a Seljuk Sultanate of Rum acute design from the Haund Hatun complex in Kayseri (1237-38), along with a variation of this design with additive swastikas from the Topkapi Scroll.Footnote 13 Like the design from Fig. 156, the fact that this example is also found at both the Huand Hatun complex in Kayseri and the Topkapi Scroll suggests a possible connection between these two historical sources. Figure 157b is a median design that can be created from either an underlying tessellation of just large octagons and pentagons, or one of small octagons, pentagons, and small hexagons. This is a Timurid design from the Bibi Khanum in Samarkand, Uzbekistan (1398-1404), and a magnificent Ilkhanid additive variation was used in the mausoleum of Uljaytu in Sultaniya, Iran (1305-1313) [Fig. 66].

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Fig. 157

Figure 158 demonstrates several secondary polygonal modules from the fourfold system A that were occasionally used in creating patterns. These are derived primarily through either truncation of the large and small octagons, or as interstice regions in tessellations of the standard polygonal modules. These additional modules were only used in the eastern regions to a limited extent, and are more commonly associated with fourfold system A designs of the Maghreb. Figure 159 illustrates a historical median pattern that incorporates one of the secondary hexagons into the underlying generative tessellation. This particularly early geometric pattern is Qarakhanid and is found on the mausoleum of Nasr ibn Ali—the middle of the three contiguous mausolea at Uzgen, Kyrgyzstan (1012-13) [Photograph 15]. This is an early date for the use of an underlying tessellation with this level of relative sophistication, and it is very likely that this example from Uzgen may have been created from the orthogonal grid method. This design was also used by Seljuk artists at the Sultan Sanjar mausoleum in Merv, Turkmenistan (1157). Figure 160 shows another Qarakhanid design that employs a secondary module within its underlying tessellation. This generative tessellation includes both the standard pentagon from this system, as well as one of the secondary pentagons from Fig. 158. The widened interweaving pattern line is arrived at through offsetting the basic pattern lines in only a single direction rather than the far more common practice of offsetting equally in both directions. This will often have a pronounced effect on the visual balance of a design. This highly successful acute design is from the anonymous southern mausoleum of the three mausolea at Uzgen (1186). Figure 161 illustrates a median pattern created from the fourfold system A that employs an underlying tessellation comprised of octagons, truncated octagons, and interstice eight-pointed stars. The unusual character of this historical example is due to their being two varieties of interweaving widened lines: those produced from offsetting the singular pattern lines in both directions equally and those that offset the pattern lines in one direction only. This unusual practice is represented in the middle panel of this figure. The single pattern line offsets are double the width of the twin offsets, thereby making all the widened lines the same thickness. This inconsistent line widening methodology results in obscuring the eight-pointed stars within the underlying octagons. This is a Zangid design from several window grilles at the Nur al-Din Bimaristan in Damascus (1154) [Photograph 37]. Figure 162 shows a Nasrid median design that utilizes two distinct rectangular repeat units, either of which can be used on its own to fill the two-dimensional plane, thereby qualifying it as a hybrid design. This example demonstrates the more complex results achieved through the use of secondary underlying polygonal modules combined with a more flexible and arbitrary approach to the application of the pattern lines to the underlying tessellation. This more refined use to the fourfold system A provided for greater artistic license that the more formulaic approach to using design systems. This approach to the fourfold system A in particular was typical in the western regions under the patronage of the Nasrids of Spain and the Marinids of Morocco. This example is from a wooden door at the Alhambra in Granada, Spain (1370) [Photograph 62]. The design in Fig. 163 is also a very successful median design with inclusions of the secondary modules from this system. Figure 163a details the more varied approach to the pattern line application typical of the Maghrebi tradition. Of particular note is the treatment within the cluster of five underlying octagons at the central periphery of the repeat. Also of note are the parallel pattern lines that create a bordering frame around the periphery of the panel. This is a common framing device in the western geometric tradition. As shown, this outer frame of parallel pattern lines typically extends beyond the line of symmetry created by the edges of the repeat unit (dashed line). Figure 163b illustrates how the underlying tessellation for this design includes multiple secondary polygonal modules, including two varieties of truncated octagon. The central 16-sided element is an interstice of the squares and smaller truncated octagons. This element allows for the central eight-pointed star that is rotated out of alignment with the rest of the design by 22.5°. This very appealing feature appears to be unique to this design. Figure 163b also illustrates how the large repeat unit of this panel is comprised of multiple repetitive elements that include a central octagon, large and small squares, half squares, rhombi, and half rhombi. This feature identifies this as a hybrid design, and all but the central octagon can be used on their own to create successful patterns. Figure 163c represents the interweaving line version of this design as per the original from the Hall of Ambassadors at the Alhambra. Figure 164 shows a Timurid median pattern with crossing pattern lines set at 67.4031…. The underlying pentagons and rhombi are secondary polygonal modules that are less typical to this system. This is from a cut-tile mosaic panel at the Shah-i Zinda funerary complex in Samarkand, Uzbekistan (1386) [Photograph 74].

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Fig. 158

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Fig. 159

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Fig. 160

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Fig. 161

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Fig. 162

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Fig. 163

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Fig. 164

The Maghreb was the only region that incorporated 16-pointed stars into designs associated with the fourfold system A. Figure 165 shows a Mudéjar median pattern with a large percentage of secondary underlying polygonal modules within the underlying polygonal tessellation, including 16-gons surrounded by elongated pentagons. These contribute significantly to the overall complexity of the design. As with other Maghrebi examples created from this system, changes to the standard pattern line application to particular polygonal modules add to the distinctive aesthetic character of this design. This design is from the Synagogue del Tránsito in Toledo (1360). The acute pattern in Fig. 166 is from a Nasrid window grille at the Alhambra. This hybrid design is comprised of two distinct repetitive elements (dashed line), either of which can be used on its own. This also employs a wide assortment of primary and secondary underlying polygonal modules, nonstandard pattern line application, and a central 16-pointed star within the upper square repeat. Figure 167 illustrates a Nasrid acute design from a zillij panel from the Alhambra. Like the previous example, this is created from primary and secondary underlying polygonal modules, nonstandard pattern line placement, and 16-pointed stars in the Maghrebi style. Figure 168 shows another acute design from a window grille at the Alhambra that employs these same methodological characteristics. This design has a particularly large repeat unit.

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Fig. 165

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Fig. 166

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Fig. 167

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Fig. 168

3.1.4 Fourfold System B

There are fewer polygonal modules that make up the fourfold system B than that of the fourfold system A. Yet even the comparatively small number will tessellate in innumerable combinations, and will produce a tremendous diversity of geometric designs. As such, the examples in this section are representative of the historical record, but only scratch the surface of the design potential offered by this methodological system. Figure 169 illustrates the five primary polygonal modules that comprise the fourfold system B, along with the standard applied pattern lines in each of the four pattern families. As with other historical systems, additional polygonal modules are occasionally employed within this system, and these are generally the product of interstice regions in tessellations made up of otherwise standard polygonal modules. Exceptions to this form of secondary module are the additional polygonal modules that allow for the incorporation of 16-pointed stars into the pattern matrix. These are very similar to the equivalent modules from the fourfold system A. The angular openings of the acute crossing pattern lines in the fourfold system B are 45°, 70.5288…° for median designs, and 112.5° for obtuse designs. There are two varieties of applied pattern line for the large hexagons of the acute family. Type A has the standard 45° crossing pattern lines located upon the midpoints of the hexagonal edges. Type B introduces a pair of parallel lines that allow for the creation of octagons within the pattern matrix as per Fig. 172. With the exception of the octagonal module, the two-point patterns created from this system generally have applied pattern lines that are parallel to the lines from the obtuse family, but set upon two points of each polygonal edge. As shown, the two-point applied pattern lines to the octagons generally employ 45° supplementary angles, while those of the other modules generally have 33.75° and 146.25° supplementary angles. There are just two edge lengths among the polygonal modules of the fourfold system B. This requires the modules to tessellate with one another in conformity with edge length. The proportions and simple constructions for each of these modules is illustrated in Fig. 170. The pentagon is easily constructed from two octagons or alternatively from the 16-gon; the rhombus and both varieties of hexagon are interstice regions created from tessellating with the octagons and pentagons.

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Fig. 169

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Fig. 170

Figure 171 demonstrates the origin of the seemingly unusual angular openings for the crossing pattern lines of the median and obtuse families. Figure 171a shows the octagonal origin for the 70.5288…° angle of the crossing pattern lines for the median family. A line that connects the midpoint of the octagon with the third sequential vertex determines this angle. Figure 171b illustrates the 112.5° angular opening of the obtuse family that results from connecting the midpoints of the short edges of the pentagon.

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Fig. 171

Figure 172 illustrates the functionality of the two varieties of acute pattern line application to the large hexagons. Figure 172a shows the standard acute pattern line application to the frequently found configuration of two large hexagons and pentagon. Figure 172b demonstrates how the addition of two parallel pattern lines results in the creation of regular octagons within the pattern matrix. This variation was widely incorporated into the acute patterns created from the fourfold system B.

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Fig. 172

Figure 173 illustrates designs from each of the four pattern families as applied to the most basic of the underlying generative tessellations produced from the fourfold system B. The dual of the underlying tessellation (dashed line) is the 4.82 semi-regular tessellation, and each of these patterns can be alternatively created from this generative tessellation. The acute pattern in Fig. 173a is the most ubiquitous design produced from this system, and examples are found throughout the Islamic world. The earliest extant example appears to be from the arch spandrel of the Seljuk mihrab in the Friday Mosque at Sin, Iran (1134). Other early locations were predominantly in Iraq and the Levant, and include the base of the Zangid minaret of the Great Mosque of Nur al-Din in Mosul, Iraq (1170-72); two side panels in the Ayyubid stone mihrab in the Zahiriyya madrasa in Aleppo, Syria (1217); a wooden soffit at the Farafra khanqah (Dayfa Khatun) in Aleppo (1237-38); and the back wall of the arched recess in the Ayyubid wooden mihrab (1245-46) of the Halawiyya mosque in Aleppo. Anatolian examples from the Sultanate of Rum include the Donar Kunbet tomb tower in Kayseri (1276); the Gök madrasa in Sivas (1270-71); the Ahi Serafettin mosque in Ankara (1289-90); and the Esrefoglu Süleyman Bey mosque in Beysehir (1297). An early Mamluk example was used in the window grilles of the Sultan Qala’un funerary complex in Cairo (1284-85) [Photograph 55]. There are also many fine examples from the eastern regions produced after the Mongol destruction. These include a cut-tile mosaic panel from the Abdulla Ansari complex in Gazargah near Herat, Afghanistan (1425-27) [Photograph 76], and a marble jali screen from the tomb of Salim Chishti at Fatehpur Sikri (1605-07) [Photograph 77]. This Mughal example has the distinction of including the underlying generative tessellation as part of the completed screen, thereby providing valuable evidence of the historicity of the polygonal technique. Somewhat surprisingly, the median pattern in Fig. 173b (by author) does not appear to have been used historically. The obtuse pattern in Fig. 173c and the two-point pattern in Fig. 173d both enjoyed limited historical use. The obtuse pattern was used by the Ayyubids in an inlaid stone panel at the Firdaws madrasa in Aleppo (1235-36), while the two-point design was used by Mamluk artists in the stone ceiling of the Ashrafiyya madrasa in Jerusalem (1482). Figure 174 shows three additional two-point designs created from the same underlying tessellation introduced in Fig. 173. All three designs are variations on the two-point pattern in Fig. 173d, and each includes a square centered on the vertex where all four pentagons meet. Figure 174a shows a Mamluk variation that is found at both the Aqbughawiyya madrasa (1340) in the al-Azhar mosque in Cairo and the Sultan al-Mu’ayyad Shaykh complex in Cairo (1415-22). Figure 174b shows a Ghurid design from the Friday Mosque at Herat, Afghanistan (1200) [Photograph 32]. Figure 174c shows a Seljuk Sultanate of Rum variation from the Bimarhane hospital in Amasya, Turkey (1308-09). Figures 174b and c are similar in their inclusion of superimposed dodecagons into their pattern matrices, but Fig. 174c differs in its arbitrary treatment of the alternating eight-pointed stars.

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Fig. 173

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Fig. 174

There are a large number of orthogonal geometric patterns created from underlying tessellations in which the octagons are separated by hexagons along the edges of the square repeat unit. A feature of this variety of underlying tessellation is the four clustered pentagons at the center of each square repeat unit. It is interesting to note that both varieties of hexagon from the fourfold system B were used within such configurations of underlying modules. Figure 175 illustrates the acute, median, and obtuse designs created from an underlying tessellation that employs the small hexagon from the fourfold system B. While fully acceptable to the aesthetics of this tradition, the acute design in Fig. 175a was not generally used. Rather, the historical acute designs associated with this configuration of underlying polygonal modules tend to employ the large hexagon as per Fig. 177a. This is a subtle difference that is not readily apparent to casual observation. The acute design in Fig. 175b is an arbitrary variation of the standard acute pattern. This extends designated lines within the pentagons to allow for an eight-pointed star at the center of each repeat unit. This variation was used in several locations in the Maghreb including: the entry portal of the Sidi Bou Medina mosque in Tlemcen, Algeria (1346); a wooden door from the Bu ‘Inaniyya madrasa in Fez (1350-55); and a zillij panel from the Alcazar in Seville (fourteenth century). The double-line treatment of this illustration conforms with the example from Tlemcen. Figure 175c is a Muzaffarid median design from the Friday Mosque at Kerman (1349). The obtuse pattern in Fig. 175d was used in several locations: by Ayyubid artists at the portal of the Palace of Malik al-Zahir at the Aleppo citadel (before 1193), by Mamluk artists at the Taybarsiyya madrasa (1309) in the al-Azhar mosque in Cairo, and by Ottoman artists in the Great Mosque at Bursa (1396-1400). Figure 176 illustrates three historical two-point patterns created from the same underlying tessellation as that of Fig. 175, and as with the 3 two-point examples in Fig. 174, each of these incorporates a square centered at the vertex of the four clustered pentagons. Figure 176a shows a Mamluk design from the Sidi Madyan mosque in Cairo (1465); Fig. 176b shows a Mamluk design from the Sultan al-Mu’ayyad Shaykh complex in Cairo (1415-22); and Fig. 176c shows a much earlier Qarakhanid example from the southern anonymous tomb among the three contiguous mausolea in Uzgen, Kyrgyzstan (1186). While following the same conceptual layout, the underlying tessellation in Fig. 177 employs the large hexagon rather than the small hexagon from the fourfold system B. As a consequence, the size and proportion of the pentagons are different from the standard pentagon from this system, and are derived by simply dividing the interstice region produced from the octagons and hexagons into four quadrants. This new interstice pentagon is not found in other historical patterns, and its unique association with this tessellation disqualifies it from inclusion with the standard modules of this system. The singular application of this pentagonal form did not preclude this tessellation from broad appeal, and indeed patterns created from this tessellation were used by several Muslim cultures. Figure 177a shows the standard acute design produced from this tessellation. Multiple examples of this design were used during the Seljuk Sultanate of Rum, and one of the earliest is from the Sultan Han in Aksaray, Turkey (1229). A roughly contemporaneous example by Ayyubid artists was used for a balustrade in the minaret of the Aqsab mosque in Damascus (1234). Figure 177b shows a variation of the acute design in Fig. 177a that effectually introduces octagons into the pattern matrix, and thereby changes the pattern lines associated with the cluster of four pentagons. This design was used by Zangid artists in the wooden mihrab of the Lower Maqam Ibrahim in the citadel of Aleppo (1168), as well as on the minbar of the al-Aqsa mosque in Jerusalem (1187). The median pattern in Fig. 177c was used by the Ilkhanids in the arch over the entry portal of the round tower in Maragha, Iran (thirteenth century). The design in Fig. 177d is unusual for its inclusion of curvilinear lines into the pattern matrix. This is a Nasrid variation from a ceiling at the Alhambra. Figure 178 illustrates three historical designs that are not created from the fourfold system B, but whose underlying tessellation shares the conceptual arrangement of octagons separated by hexagons along each edge of the square repeat unit. Because of this conceptual similarity, these designs are included within this section. The hexagon in this tessellation is regular, and the four resulting interstitial pentagons are proportioned accordingly. Figure 178a shows a Mamluk acute design from the Vizier al-Salih Tala’i mosque in Cairo (1303). Figure 178b shows a Mamluk median pattern from a carved stone relief panel at the Altinbugha mosque in Aleppo (1318). And Fig. 178c shows an Ildegizid median design from the façade of the Mu’mine Khatun mausoleum in Nakhichevan, Azerbaijan (1186).

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Fig. 175

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Fig. 178

As with the fourfold system A, geometric patterns that employ the hybrid repetitive structure of the 4.82 grid were also produced from the fourfold system B. Figure 179 illustrates two historical examples of this variety of design. Figure 179a shows an acute pattern that was relatively well known throughout Muslim cultures. The rings of eight octagons that are centered on each vertex of the square repeat unit result from the utilization of the type B large hexagon [Fig. 169]. This is a distinct visual characteristic of all designs that include this feature. It is interesting to note that the design within just the repetitive square region is identical to that of Fig. 173a. The earliest use of this design appears to date from the Seljuk Sultanate of Rum and is found at the Izzeddin Keykavus hospital and mausoleum in Sivas (1217-18). Gerd Schneider’s excellent research on the Islamic geometric patterns of the Seljuk Sultanate of Rum identified no less than ten examples of this design produced in Anatolia during this period.Footnote 14 A Kartid example of this design was used in the painted fresco ornament of the mausoleum of Shaykh Ahmed-i Jam at Torbat-i Jam in northeastern Iran (1442-45) [Photograph 75]. There are also several Mamluk examples of this design, including the side panel of a minbar (1296) for the ibn Tulun mosque; a Quranic Illumination from HebronFootnote 15 (1369); and an Armenian katshkerim carved stone relief panel set into the exterior walls of the Cathedral of St. James in Jerusalem (thirteenth to fourteenth centuries). It is worth noting that during the period of the Mamluks and the Seljuk Sultanate of Rum, the Armenian Christians occasionally adopted Islamic geometric design into their carved stone decoration in both Armenia and, to a lesser extent, in Jerusalem. Like the thirteenth- and fourteenth-century Coptic Christians in Cairo and the Catholics and Jews in al-Andalus, the Armenian Church was not averse to adopting the prevailing geometric aesthetics of their Muslim neighbors. The obtuse pattern in Fig. 179b is produced from the same underlying tessellation. This was created during the Seljuk Sultanate of Rum and is found at the Hudavent tomb in Nidge (1312). Figure 180 illustrates two further designs created from the same underlying tessellation set within the 4.82 hybrid repetitive structure. Figure 180a shows a median design (by author), and Fig. 180b shows a two-point design (by author). These two examples demonstrate the efficacy of this system in creating original patterns that adhere to the aesthetics of this ornamental tradition.

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Fig. 180

As with the fourfold system A, several examples of patterns that employ the rhombic repeat unit with 45° and 135° included angles were produced from the fourfold system B. Figure 181 illustrates an acute pattern that uses this repeat. This pattern includes the type B large hexagons within its underlying tessellation. This produces the distinctive bands of zigzag octagons throughout the pattern matrix. The earliest example of this pattern appears to date from the Seljuk Sultanate of Rum: located at the Izzeddin Keykavus hospital and mausoleum in Sivas, Turkey (1217-18). Artists working under the auspices of this dynasty included multiple subsequent examples of this design in their work.Footnote 16 A later Timurid example of this design comes from the carved stucco ornament of the mausoleum of Amir Burunduq in the Shah-i Zinda funerary complex in Samarkand, Uzbekistan (1390). Figure 182 illustrates an acute design that utilizes two separate hybrid repeat units (dashed lines) within its overall rectangular repeat: a square and rhombus. On its own, the design within the square repeat is identical to that of Fig. 173a; and the design within the rhombic repeat is the same as the design in Fig. 181. These work together by virtue of the identical edge configuration in the underlying tessellation between that of the square and the short edge of the half rhombic triangle. This ingenious hybrid design is the work of Ildegizid artists working on the Mu’mine Khatun mausoleum in Nakhichevan, Azerbaijan (1186), and a later example is found at the Abbasid Palace of the Qal’a in Baghdad (c. 1230) [Photograph 34]. It is interesting to note that this twelfth-century use of the rhombic component of this hybrid design predates the earliest known use of this rhombus as a single repeat unit at the Izzeddin Keykavus hospital and mausoleum by some 32 years. Figure 183 shows a two-point pattern that also uses the rhombic repeat with 45° and 135° included angles. In addition to octagons, small hexagons, and pentagons, the underlying tessellation for this design includes the rhombic module. The use of this module is relatively unusual. This design was used by Mamluk artists in two carved stone panels above the mihrab at the al-Mar’a mosque in Cairo (1468).

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The historical use of the fourfold system B was less inclined to include secondary interstice elements within the underlying tessellations as the fourfold system A. However, the few noteworthy examples help to emphasize the design flexibility of systematic methodologies generally, and the potential for interesting original designs that can still be created using the fourfold system B. Figure 184 illustrates an obtuse pattern created from an underlying tessellation comprised of octagons, pentagons, large hexagons, and two interstice elements: squares and rhombi. This rhombus has the same included angles as the standard rhombus module [Fig. 169], but the edge length corresponds to the shorter edges of the polygonal modules rather than to the longer edge. Another unusual feature of this design is the variable angular openings of the crossing pattern lines. In addition to the standard 112.5° of the obtuse family, this example includes 90° and 104.1776 …° angled crossing pattern lines. This latter angular opening is a product of incorporating regular octagons into the region of extended pattern lines from the obtuse eight-pointed stars. This is a Mamluk design from a stone lintel at the Sultan Qaytbay Sabil in Jerusalem (1482).

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Fig. 184

Patterns that incorporate 16-pointed stars are less common to the fourfold system B than to the fourfold system A. However, those that were produced are very successful, and much like those of their fourfold system A counterparts. Figure 185 illustrates two designs created from the same underlying tessellation with the additional secondary polygonal modules that allow for the incorporation of 16-pointed stars into the pattern matrix. These designs employ the same 4.82 hybrid repetitive structure as the examples from Figs. 179 and 180 (dashed lines). The 16-fold ring of pentagons within the underlying tessellation have variable proportions, with those in the two-point pattern being closer to the proportions of a regular pentagon. Figure 185a shows a Nasrid acute pattern from the zillij mosaics at the Alhambra [Photograph 63]. This was used subsequently at the Sa’dian tombs in Marrakesh, Morocco (sixteenth century), and in an Alawid stucco ceiling of the Moulay Ishmail mausoleum in Meknès, Morocco (seventeenth century). Figure 185b shows a two-point design from the Imam al-Shafi’i mausoleum in Cairo (1211). This is a rare example of a fourfold systematic design with incorporated 16-pointed stars from a region other than the Maghreb. Figure 186 shows an acute design created from this system that incorporates 8- and 16-pointed stars into a pattern matrix that repeats with both a rhombic grid (dash lines) and hexagonal grid (white). The artist who created this design resolved the pattern lines associated with the disproportionately long pentagons that separate the 16-gons in the underlying tessellation very nicely. This is a Marinid design from a wooden ceiling at the Bu Inaniyya madrasa in Fes, Morocco (1350-55) [Photograph 64].

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3.1.5 Fivefold System

The fivefold system is immensely important to the history of Islamic art and architecture. More than any of the other systematic design methodologies, the fivefold system received significant innovations in pattern line variations, ever-greater design complexity, and repetitive geometric structures. This was also the most broadly dispersed methodological design system throughout the Islamic world, with the greatest diversity and the largest number of representative examples. The recognizable qualities of patterns created from this system were a significant contributing factor to the cohesive aesthetic of Islamic geometric art throughout the length and breadth of this tradition. The fivefold system has significantly more polygonal modules than the previously discussed design systems, and the greater the number of polygonal components, the greater the diversity of tessellating potential within a given system. Even with the large number of historical designs created from this system, there is still tremendous potential for creating original patterns. This extends to all of the varieties of design that the fivefold system was applied to, including field patterns; designs that repeat with a single rhombic, rectangular, or hexagonal repetitive cell; patterns that employ hybrid repetitive stratagems with more than a single repetitive cell; designs that transition between more than one scale of polygonal module; designs that incorporate 20-pointed stars into the pattern matrix; and dual-level designs with self-similar characteristics.

The angular openings of the crossing pattern lines in the fivefold system all relate directly to the geometry of the decagon. Figure 187 demonstrates how the angular openings of the acute family are 36°, those of the median family are 72°, and those of the obtuse family are 108°. By convention, the applied pattern lines of the two-point family follow the angles of the obtuse family, but allied to two points on each polygonal edge rather than the midpoint. The supplementary angles are therefore 36° and 144°. The application of the angles associated with each pattern family is more standardized than in the system of regular polygons, as well as in both fourfold systems. Yet there are significantly more conventions for arbitrary design modification—sometime additive, and sometimes subtractive—that were applied to the fivefold system, and led to even greater overall design diversity.

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Fig. 187

As with both fourfold systems, there are two edge lengths in the polygonal modules that comprise the fivefold system. The individual polygons in this system may have just the short or long edge lengths, or may be a combination of both. These edge lengths have a φ (phi) proportional relationship, and indeed the golden ratio is inherent to the fivefold system. Figure 187 illustrates the modules from this system that have only the short edge length. Being that the fundamental module to this system is the decagon, the shorter edge length of these modules can be regarded as the primary edge length of this system. This figure also demonstrates the standard pattern lines in each of the four pattern families applied to each of these modules. The visual character of the applied pattern lines of certain modules (shaded) within specific pattern families is less acceptable to the aesthetics of this tradition. Specifically, the acute pattern lines created from the concave and long hexagons are generally less acceptable design features within this system and are rarely found in historical examples. Figure 188 provides the additional polygonal modules that have both edge lengths, as well as those that have only the long edge lengths. The median pattern lines applied to the two conjoined triangular modules (1/5 of the dodecagon) that make the quadrangular kite shape are occasionally given the specialized treatment shown in this figure. The shading over the triangles in the obtuse and two-point families indicates that these two varieties of pattern line application do not make successful designs. With the exception of the large pentagon, large rhombus and large concave hexagon in the acute family, the large pentagons, large rhombi, and the large concave hexagons were rarely used historically. When they were, there is considerable variation in their pattern line application, and the examples shown are only representative.

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Fig. 188

Figure 189 demonstrates the simplicity of constructing the more common polygonal modules of this system from the dodecagon. The short edge length equals the sides of the decagon and the long edge equals the radius of the decagon, and as stated these have a φ proportional relationship. Figure 190 demonstrates how some modules of the fivefold system can also be produced as interstice regions when tessellating with the decagon and pentagon. Figure 190a demonstrates the concave hexagon as a product of four decagons in a rhombic arrangement and Fig. 190b demonstrates how the barrel hexagon can be derived from an arrangement of decagons and pentagons. Figure 191 demonstrates the creation of new modules through decagonal mirroring and decagonal union. Figure 191a illustrates how the thin rhombus, the long hexagon, and the octagon can be created from reflected decagons. These three examples also show how each can be produced as interstice regions of a tessellation of other modules from the system. Figure 191a further illustrates how the wide rhombus can be created from two superimposed pentagons. Figure 191b demonstrates how the union of decagons can create new larger modules: in this case, three types of conjoined decagons. This figure also shows how the barrel hexagon can be created from superimposed pentagons. Patterns created from the larger conjoined decagons are relatively unusual. As geometric figures these twinned decagons conform to Johannes Kepler’s “fused decagon pairs” or “monsters,”Footnote 17 although their earliest use as a design vehicle predates Kepler by over a century. Figure 192 provides examples of pattern line application to two varieties of conjoined decagons in each of the four pattern families. The conjoined decagons in Fig. 192a have three edges of superimposition, while those in Fig. 192b have two. The application of pattern lines to larger polygonal modules such as these is open to greater artistic license: for example, the rosette infill of the obtuse and two-point pattern lines in Fig. 192a is an arbitrarily derived motif rather than purely the product of the pattern lines as it relates to the polygonal edge conditions (as per the obtuse and two-point pattern lines in Fig. 192b). Figure 193 shows several additional examples of decagonal truncations with their applied pattern lines. It is worth noting that the applied pattern lines in each of the four families do not necessarily work well with every truncation. The pattern lines in the examples shown in this figure have a synergistic relationship between the midpoints of the decagonal edges and the longer truncated edges. Truncated decagons were rarely employed traditionally, but offer a further range of potentiality for contemporary designers. Figure 194 illustrates multiple tessellating configurations that employ truncated decagons. The upper eight examples are radial, and the lower four are linear. These varieties of radial and linear combinations of truncated decagons are generally ahistorical, and are included because they nonetheless have valid potential for creating contemporary designs that fall within the aesthetics of this design tradition.Footnote 18 Figure 195 demonstrates the φ proportional relationship between the short and long edges of several representative polygonal modules that make up the fivefold system, including the decagon and pentagon. These golden ratio proportions are an inherent aspect of both the underlying polygonal tessellations, and their applied pattern lines.

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Figure 196 illustrates eight examples of tessellating configurations that employ the triangle module from the fivefold system. This triangle is 1/10 of the decagon, and the examples illustrated can be thought of as partial decagons. These are analogous to the use of the assembled 1/8 triangular modules from the fourfold system A [Fig. 133]. As shown in this figure, the historical use of contiguous 1/10 triangles in a rotational assemblage is almost always associated with acute patterns. While very successful with the applied pattern lines in this family, these arrangements are generally less successful when applied to the other three pattern families. Of particular note are the pattern elements that are created at the acute triangular vertices. With the exception of the two 7/10 arrangements, these are relatively common features of more complex acute patterns created from this system [Figs. 254–256 and Fig. 266]. The features of the 8/10 arrangement are the same as that of the acute example in the conjoined decagons of Fig. 192b.

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Fig. 196

The fivefold system has greater nuance than either of the fourfold systems. Just as there are fivefold polygonal modules that are less acceptable to the applied pattern lines of specific pattern families [Figs. 187 and 188], it is also important to note that the standard pattern line applications do not always work well in all modular configurations. Figure 197 demonstrates the pattern lines in each of the four pattern families applied to an assembly of six pentagons surrounding a thin rhombus. This configuration can be seen to create very successful motifs within the obtuse and two-point families, but unacceptable features in both the acute and median families. The problems with the pattern lines within these two latter families are the multiple small background elements, creating constrained regions that are out of balance with the rest of the design, and outside the aesthetic expectations of this ornamental tradition. In such cases, the design can be remedied through either changing the polygonal modules, or adjusting the pattern lines within the existing tessellation. Figure 198 demonstrates two methods of correcting the unacceptable acute pattern line conditions that result from this cluster of pentagons surrounding the thin rhombus illustrated in Fig. 197. The first of these replaces the four pentagons that are edge-to-edge with the thin rhombus with four trapezoids. This provides for the use of the large wide rhombus within the newly created interstice region. The second replaces all six pentagons with six trapezoids, and fills the void with the large concave hexagon. Each of these are very successful corrections, and well known to the historical record. Figure 199 illustrates the same corrective measures within the median family. The solution that employs four trapezoids is not particularly attractive and, not surprisingly, is absent from the historical record. By contrast, the solution with six trapezoids is aesthetically pleasing and, indeed, found with some frequency within the historical record. Figure 199 also demonstrates how corrections to the median pattern lines can be achieved through a subtractive modification of the crossing pattern lines that are shared by both the pentagons and rhombus. There is also historical precedent for this solution.

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Figure 200 demonstrates how underlying polygonal tessellations created from the fivefold system will often have a dual relationship with another tessellation also comprised of polygonal modules from this same system. This is analogous to many of the underlying tessellations of the fourfold system A [Fig. 134]. Both tessellations in Fig. 200 will create patterns in each of the four pattern families, and frequently a single geometric pattern can be created from either with equal ease. It is therefore not always possible to know with certainty the underlying polygonal tessellation that was used to create a specific historical example. The dual relationship between the tessellations in Fig. 200a and c is demonstrated in Fig. 200b. As said, both of these tessellations will produce a large number of historical designs with equal facility. Figure 201 illustrates the application of pattern lines in all four pattern families to this tessellation. The pattern in Fig. 201a is the classic acute design from the fivefold system. This is characterized by 36° angular openings at the midpoints of the polygonal edges of the tessellation comprised of decagons, pentagons and barrel hexagons. When examined from the perspective of the generative tessellation of decagons and concave hexagons, the angular opening for this design is 144°. However, this change in the angular opening does not change the categorization of the pattern family. The fivefold acute family is identified by the aesthetic character of the design wherein the five- and ten-pointed stars are governed by 36° points, regardless of the underlying tessellation that was used to create them. Another more favorable feature of the underlying tessellation of decagons, pentagons, and barrel hexagons is the fact that the ten-pointed stars are directly generated from the underlying tessellation. This is not the case with the tessellation of decagons and concave hexagons, wherein the ten-pointed stars are produced through an additive process. This limitation of the underlying tessellation of just decagons and concave hexagons is also the case with the median pattern from Fig. 201b. This design has 72° angular openings at the midpoints of the decagons, pentagons, and barrel hexagons, and 108° angular openings at the midpoints of the dual tessellation of decagons and concave hexagons. The 108° angular openings are standard to the obtuse pattern family, and would suggest that this design can be regarded as either median or obtuse. However, this example is rightfully regarded as a median pattern due to the distinctive aesthetic quality of the five-pointed stars with 72° points. The pattern in Fig. 201c is similarly characterized as an obtuse pattern due to the strong aesthetic character of the pentagonal pattern lines with 108° angular openings placed within the underlying pentagons of the tessellation. This is despite the 72° angular openings of the pattern lines in the dual tessellation that would otherwise define it as a median pattern. The two-point pattern in Fig. 201d has the distinctive two-point characteristics in both generative tessellations, albeit with differing supplementary angles along the polygonal edges.

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Fig. 201

A key methodological criterion of the polygonal technique relies upon the population of each edge of the pattern’s repeat unit with a specific configuration of polygonal modules, and associated pattern lines, so that they match the opposite edge of the repeat unit when this is applied to the plane through translation symmetry. As a general rule, the greater the number of polygonal modules that rest upon the edges of the repeat unit the greater the amount of geometric information contained within each repeat unit, and the greater degree of complexity in the resulting pattern. Figure 202 illustrates a number of typical examples of polygonal configurations for application to the edges of repeat units. In each case decagons are placed at the ends of each line representing the edge of the repeat unit. Arrangements such as these are equally applicable to rhombic, rectangular, and hexagonal repeat units, and—it is important to note—are also used in constructing the secondary pattern layout for dual-level designs. Figure 203 demonstrates the step-by-step application of two very basic polygonal edge configurations to rhombic and rectangular repeat units. The upper row employs the edge configuration of two decagons separated by two edge-to-edge pentagons to produce a rhombic repeat unit with 72° and 108° included angles. The central interstice region is conveniently the barrel hexagon from the fivefold system. The next row uses the same edge configuration for the short edge of a rectangular repeat unit and two decagons separated by the barrel hexagon for the long edge. The central interstice region is once again easily filled with modules from the fivefold system. The third row uses two decagons separated by a barrel hexagon as the edge configuration of a rhombic repeat unit with the 72° and 108° included angles, with an infill of pentagons and a thin rhombus in the central interstice region. And the bottom row uses this same edge configuration for a rhombic repeat unit with 36° and 144° included angles, with two wide rhombi filling the remaining interstice regions. Each of these four repeat units produces patterns that are well known to the historical record. Figure 204 illustrates a step-by-step construction of a more complex edge configuration for a rhombic repeat unit with 72° and 108° included angles. The more polygonal information along each edge of the repeat unit, the larger the resulting interstice region; and the larger the interstice regions, the more options for secondary polygonal infill. The final image in this figure suggests an infill with conjoined decagons, and this can be a very successful solution to populating the interstice region, but other options are possible. Figure 205 shows eight alternative polygonal arrangements for populating this interstice region. Figure 205a illustrates four infill treatments that have two axis or reflected symmetry. This is typical, but not mandatory with this tradition, and each of these infill configurations will create significantly different geometric patterns. By contrast, the polygonal infill configurations in Fig. 205b have twofold rotational point symmetry. While rare, this is occasionally found within the historical record [Fig. 266].

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Interstice regions within generative tessellations will often create design challenges that engender creative solutions. A case in point is a fivefold pattern from the Ghaznavid minaret of Mas’ud III in Ghazni, Afghanistan (1099-1115).Footnote 19 Figure 206 shows how this design is derived from a tessellation of corner-to-corner touching decagons placed upon a rhombic grid with concave decagonal interstice regions separating the decagons. Atypically, the decagonal tessellation is maintained as part of the completed design. This exceptional fivefold pattern originates from the period when fivefold patterns were first being introduced into Islamic ornament, and the use of this decagonal tessellation is more experimental, and less akin to the standard conventions of the fivefold system. This experimental approach is seen in the non-regular seven-pointed stars that result from the treatment of the pattern lines within the interstice region.

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Fig. 206

In contrast to both fourfold systems, the greater number of secondary polygonal modules within the fivefold system provides for their significantly greater design potential in creating field patterns. These exclude the decagon within their underlying polygonal structures, and hence lack the ten-pointed stars that are otherwise a standard feature of designs produced from this system. The earliest fivefold field pattern appears to be Seljuk: used in the interior ornament of the Khwaja Atabek mausoleum in Kerman (1100-1150) [Fig. 211]. Indeed, this variety of fivefold pattern was especially popular among Seljuk artists, with the greatest number being used as border designs during the Seljuk Sultanate of Rum. The absence of decagons in the underlying tessellations of fivefold field patterns generally precludes the use of rhombic repeat units, and field patterns created from the fivefold system most commonly repeat upon either a rectangular or hexagonal grid. Figure 207 shows a two-point field pattern from the Great Mosque at Malatya in Turkey (1237-38) that repeats upon a rectangular grid. This is an unusual example in that the majority of fivefold field patterns are either of the median or obtuse pattern families. The underlying generative tessellation that produces this design is comprised of pentagons, barrel hexagons, and long hexagons. Figure 208 represents an obtuse field pattern from the Haci Kilic madrasa in Kayseri, Turkey (1275). This repeats upon a rectangular grid and the underlying generative tessellation is made up of pentagons, barrel hexagons, long hexagons, wide rhombi, and thin rhombi. This historical example utilizes only the highlighted isolated linear region as a border design. Figure 209 shows a median field pattern from the hospital (1205) associated with the Çifte madrasa in Kayseri, Turkey. This repeats upon a rectangular grid, and is created from an underlying generative tessellation comprised of concave hexagons, trapezoids that are half-concave hexagons, and wide rhombi. The short edges of the four clustered trapezoids define the small thin rhombus that is not typical to the fivefold system. Anomalies such as this abound in the application of the fivefold system, and this cluster of four half-concave hexagons around a small thin rhombus is an unusual tessellating condition that works very well in the median family: creating the distinctive 12-sided motif located at the vertex of each rectangular repeat. Figure 210 shows a field pattern from the Huand Hatun complex in Kayseri, Turkey (1237), that has an unusually broad rectangular repeat. The underlying generative tessellation has concave hexagons, long hexagons, and wide rhombi placed in fivefold rotational symmetry. Figure 211 shows the field pattern from the above-mentioned Khwaja Atabek mausoleum in Kerman, Iran, and later at the Sahib Ata mosque in Konya, Turkey (1258), and the minbar of the Esrefoglu mosque in Beysehir, Turkey (1296-97). This median pattern repeats rectilinearly, although it simultaneously repeats with a 6-sided chevron devise [Fig. 20]. The underlying generative tessellation is comprised of kite-shaped quadrilaterals with bilateral symmetry, each made up of two adjacent 1/10 decagonal triangles from this system [Fig. 188]. As repetitive elements, these quadrilaterals alternate in 180° rotation to fill the plane. This quadrangular module was used with some frequency in median patterns. However, this design is unique in that the underlying generative tessellation is comprised solely of this module.

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Fig. 211

Most historical examples of fivefold field patterns repeat upon a hexagonal grid. Figure 212 shows a Mamluk median design from the Amir Ghanim al-Bahlawan funerary complex in Cairo (1478). The underlying generative tessellation utilizes the same arrangement of linearly alternating quadrilaterals as the field pattern from the Khwaja Atabek mausoleum in Kerman, but also includes an adjacent linear band of edge-to-edge concave hexagons separating the quadrilaterals. Figure 213b shows an obtuse field pattern found in several locations in Anatolia, including Sitte Melik tomb in Divrigi, Turkey (1196); the Alaeddin mosque in Konya, Turkey (1219-21); and the Mama Hatun tomb tower in Tercan, Turkey (thirteenth century). This design can be produced with equal facility by either of the two dual grids. Figure 213a employs pentagons, barrel hexagons, and thin rhombi, and the angular opening of the crossing pattern lines is 108°: the angle associated with obtuse patterns. The alternative underlying generative tessellation in Fig. 213c employs concave hexagons and long hexagons, with 72° angular openings. This angle is a standard feature of median patterns. However, as with other patterns with these characteristics, this design is regarded as an obtuse pattern due to the visually dominant pentagonal motif. Figure 214 illustrates two field patterns created from the same underlying tessellation of just pentagons and long hexagons. Figure 214a shows a median design from the Alaeddin mosque in Konya (1219-21). As highlighted, this was used as a border design. Figure 214c shows an obtuse design from the Çifte Minare madrasa in Erzurum (late thirteenth century) that was also used as a border. Figure 215 illustrates a median field pattern also from the Çifte Minare madrasa in Erzurum. The underlying generative tessellation of this pattern incorporates the same layout of four half-concave hexagons placed around a small thin rhombus as found in the previously cited example from the hospital in Kayseri (Fig. 209). Figure 216 shows a median field pattern from the Sultan Han at Kayseri, Turkey (1232-36). The underlying generative tessellation utilizes the same kite shaped quadrilateral modules, with the same median pattern line application, as the patterns in Figs. 211 and 212. The median pattern in Fig. 217 is from the exterior cut-tile mosaic façade of the Great Mosque in Malatya, Turkey (1237-38). The underlying tessellation in Fig. 217a incorporates the cluster of four trapezoids (half-concave hexagons) that are edge to edge with a small thin rhombus: a feature of Figs. 209 and 215. The median pattern in Fig. 218 also incorporates this underlying configuration. This contemporaneous design is from the Huand Hatun in Kayseri (1237), and has an unusually long hexagonal repeat unit. The underlying generative tessellation of this median field pattern utilizes the same arrangement of four half-concave hexagons placed around a small thin rhombus seen in Figs. 209, 215, and 217, but separates these with alternating kite-shaped quadrangles as per Figs. 211, 212, and 216. The median pattern in Fig. 219 is from the Hekim Bey mosque in Konya (1270-80). This is created from an underlying tessellation of pentagons, long hexagons, and wide rhombi. Figure 220b shows an obtuse field pattern from the Great Mosque in Malatya (1237-38). This design was also used by Chaghatayid artists at the Bayan Quli Khan mausoleum in Bukhara (1357-58). As typical with this family of pattern, it can be created from either of two underlying tessellations. Figure 220a employs pentagons, barrel hexagons, and thin rhombi, and the angular opening of the crossing pattern lines is 108°—the angle associated with obtuse patterns. The alternative underlying generative tessellation in Fig. 220c employs concave hexagons and long hexagons, with 72° angular openings—the angle associated with median patterns. However, for purposes of clarity of identification, the visually dominant pentagonal motif, with its 108° angles, identifies this as an obtuse pattern.

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Fig. 212

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Fig. 213

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Fig. 214

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Fig. 215

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Fig. 216

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Fig. 217

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Fig. 218

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Fig. 219

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Fig. 220

3.1.6 Fivefold System: Pattern Variation and Modification

There is considerable historical variation in the application of pattern lines to the decagons in the fivefold system. Figure 221 illustrates standard pattern line variations in each of the four pattern families as applied to the decagon. The decision as to what form of primary ten-pointed star to incorporate into a given design is a balance between an artist’s and potentially client’s personal predilections, regional and cultural design conventions, and constraints of the end material over the design process. Generally, the acute and median pattern lines that are applied to the underlying decagon conform to the basic alternatives offered in this illustration. By contrast, the decagons of the obtuse and two-point families lend themselves to a greater amount of stylistic variation that is not governed by the inherent geometry of the underlying decagon itself. The most common variety of arbitrary additive elaboration of obtuse and two-point patterns introduces a tenfold star rosette into the otherwise open ten-pointed stars. Figure 222 demonstrates a simple process for creating the tenfold geometric rosette within both the obtuse and two-point families. This involves the following: (A) draw a radius from the center of the decagon to the midpoint of an edge; (B) extend pattern lines that are perpendicular to the radius as shown; (C) bisect angle and draw a circle as shown; (D) copy and rotate the bisected angle to the decagon’s radius; (E) trim the dart motif, draw two new radii as shown, and extend the rotated pattern lines to the new radii; and (F) copy and rotate this motif ten times around the center of the decagon.

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Fig. 221

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Fig. 222

Subtractive processes can also be applied to the median crossing pattern lines of the standard ten-pointed star created from the decagon surrounded by pentagons and barrel hexagons. Figure 223 demonstrates the most widely used variety of median pattern modification—both among patterns created from the fivefold system as well as with nonsystematic designs. This form of modification to the primary stars was especially popular with the Mamluk artists of Egypt. Figures 223a through d illustrate how the pattern lines can be removed, and replaced with a central rosette comprised of a ten-pointed star and ten darts. This motif is typically associated with fivefold patterns in the acute family. This process involves the removal of the 72° crossing pattern lines located on the decagonal edges in Fig. 223b; the extension of the remaining pattern lines in Fig. 223c; and the incorporation of a central ten-pointed star rosette as per Fig. 223d. Figure 223e shows the standard median pattern without this modification, and Fig. 223f shows this same pattern with the modification. The application of this modification to each decagonal region radically transforms the character of the original, and the replacement of the five-pointed stars with two darts and two shield shapes provides this modified pattern with the characteristics of the acute family. In fact, Fig. 223g shows how the modified design from Fig. 223f is identical to an acute pattern that is more efficiently produced from its own underlying tessellation comprised of decagons, barrel hexagons, trapezoids, and large concave hexagons. This alternative method for creating the same design makes use of the corrective adjustment to the underlying tessellation that employs six trapezoids demonstrated in Fig. 198.

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Fig. 223

Two varieties of obtuse modification to the ten-pointed stars created from the decagon are illustrated in Fig. 224. This modification was popular among Seljuk artists in both Persia and Anatolia. Figure 224a transforms the region of the decagon from a ten-pointed star into a fivefold motif with a pentagon at the center of the decagon. The less common modification in Fig. 224b replaces the ten-pointed star with a five-pointed star with 72° points as per the median family. When applied to every ten-pointed star within a given design, each of these modifications transforms the design into a field pattern.

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Fig. 224

There are multiple conventions for the arbitrary treatment of the ten-pointed stars in the two-point family. Figure 225 illustrates three examples. Figure 225a shows the standard two-point design with its mirrored pattern lines along each decagonal edge. Figure 225b extends, rather than mirrors, the lines within the pentagons to create the tenfold geometric rosette. Figure 225c extends the standard pattern lines of Fig. 225a into a tenfold geometric rosette of smaller scale than that of Fig. 225b. And Fig. 225d mirrors two of the lines within each underlying pentagon to create a distinctive ring of ten rhombi. Each of these variations was used historically.

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Fig. 225

As applied to otherwise complete patterns, arbitrary modifications to the standard pattern lines can have a pronounced effect on the visual character of a geometric design. Pattern modification is, therefore, an extremely important, if frequently overlooked, aspect of this ornamental tradition. The conventions for pattern modification are often regional and cultural, and find their most expansive representations within designs created from the fivefold system. What is more, specific pattern modifications established within the fivefold system almost certainly served as analogs for similar application to designs created from other systematic and nonsystematic expressions of the polygonal technique. As mentioned above, acute patterns created from the fivefold system received less variational attention than those produced from the other pattern families. Figure 226d illustrates and outstanding exception to this general rule. This Ilkhanid example is found at the mausoleum of Uljaytu in Sultaniya (1307-13), and is based upon the classic acute pattern created from the underlying tessellation of decagons, pentagons and barrel hexagons placed upon a rhombic repeat shown in Fig. 226a. The equal width of the two colored foreground and background elements in this design from Sultaniya are akin to the ablaq ornament of the Ayyubids and Mamluks. The standard acute pattern in Fig. 226b is characterized by the 36° angular openings of the pattern lines placed at the midpoints of the underlying polygonal edges. Figure 226c illustrates this frequently encountered interpretation with widened interweaving lines, and an early example of this classic fivefold pattern was used on the intrados of the Ghurid ceremonial arch at Bust (1149) [Photograph 33]. Seljuk artists used a widened line version as a border within the iwan of the Friday Mosque at Gonabad (1212) [Photograph 23].

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Fig. 226

As referenced, there are a considerable number of additive and subtractive historical modifications that were applied to the pattern lines of the median family. The three designs in Fig. 227 illustrate a progressive modification of the standard median pattern created from the rhombic tessellation of decagons, pentagons and barrel hexagons. Figure 227a demonstrates the standard application of the median pattern lines to this underlying tessellation. As mentioned previously, this is characterized by 72° angular openings of the pattern lines at each underlying polygonal midpoint. Figure 227b represents the commonly found tiled interpretation of this design, including in the exterior cut-tile mosaic of the Bibi Khanum mosque in Samarkand (1399-1405). Widened line interpretations of this design were also well known and include the main façade of the Sultan al-Ghuri madrasa in Cairo (1503-05). This standard line treatment has the open center within the ten-pointed stars. Figures 227c and d illustrate the historical variation to the ten-pointed stars that is relatively common among median patterns [Fig. 223d]. This example arbitrarily applies this modification to alternating decagonal regions, thereby creating a ten-pointed star rosette at just the center of the design, and not at the corners. Historical examples with this alternating modification are uncommon, but this was used by Qara Qoyunlu artists in a cut-tile mosaic border at the Imamzada Darb-i Imam in Isfahan (1453). Figures 227e and f apply this same modification to each underlying decagonal region. This has the effect of eliminating two of the points from each of the five-pointed stars in the original design. This modification maintains only a tenuous association with the underlying tessellation, effectively transforming the standard median pattern into an acute pattern that is well known to the historical record. Indeed, this same design can be produced more efficiently from either an underlying tessellation of edge-to-edge decagons [Fig. 232f] or underlying decagons that are separated by barrel hexagons [Fig. 232h]. This is a good example of both the occasional interchangeability between pattern families, and the methodological uncertainty that is often associated with specific geometric designs. Early examples of this design are found on the exterior façade of the Ildegizid mausoleum of Mu’mine Khatun in Nakhichevan, Azerbaijan (1186), and on the Zangid entry door at the Awn al-Din Meshhad in Mosul, Iraq (1248). This design was particularly popular among Mamluk artists in Cairo, and examples are found at the Sultan Qala’un funerary complex in Cairo (1284-85), the Amir Sanqur al-Sa’di funerary complex in Cairo (1315), the Hasan Sadaqah mausoleum in Cairo (1315-21), the Sultan Qaytbay funerary complex in Cairo (1472-74), the Qadi Abu Bakr Muzhir complex in Cairo (1479-80), and the Amir Azbek al-Yusufi complex in Cairo (1494-95) [Photograph 46]. The process of removing two points from each of the five-pointed stars in Fig. 227e can be replicated with the removal of other combinations of points. Each of the designs in Fig. 228 truncates two of the points from the five-pointed stars of the standard median pattern in Fig. 227a, thereby removing two of the five 72° crossing pattern lines from the midpoints of the underlying pentagonal edges. This modification in Fig. 228a is the same as that presented in Fig. 227e, with a widened line treatment in Fig. 228b. Figure 228c truncates two alternative points from the five-pointed stars: those associated with the edge-to-edge pentagons, but not the shared edges of the pentagons and barrel hexagons. The resulting interweaving pattern in Fig. 228d was used in the late Abbasid mausoleum of ‘Umar al-Suhrawardi in Baghdad (early thirteenth century). Figures 228e and f remove yet another pair of the 72° crossing pattern lines. While certainly acceptable to the aesthetics of this tradition, somewhat surprisingly, this design (by author) does not appear to be historical.

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Fig. 227

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Fig. 228

Arbitrary modifications to fivefold obtuse patterns are also relatively common. Figure 229a illustrates the standard obtuse pattern along with its underlying generative tessellation of decagons, pentagons and barrel hexagons. Figure 229b shows an interweaving line version of this very-well-known standard pattern. This design is 1 of the 3 fivefold examples that were used in the Seljuk northeast dome chamber in the Friday Mosque at Isfahan (1088-89) [Photograph 21], and is, as such, one of the earliest extant fivefold patterns known to the historical record. This example from Isfahan is the earliest known design to include the ten-pointed star rosettes within each of the underlying decagonal regions [Fig. 221]. Figures 229c and d represent a historical example with the arbitrary additive treatment applied to alternating ten-pointed stars. This modification introduces a pentagonal motif within the ten-pointed stars in the fashion illustrated in Fig. 224a. This design with alternating decagonal infill was used in the frontispiece of a Mamluk Quran (1369) commissioned by Sultan Sha’ban (r. 1363-77). Figures 229e and f represent the classic obtuse pattern with the same additive modification applied to each of the ten-pointed stars. As mentioned, the application of the modification to each ten-pointed star has the effect of transforming the pattern into a field pattern. This modified pattern is found at the Haci Kilic madrasa in Kayseri (1275). Figure 230 shows a variation of the classic obtuse design created from this same underlying tessellation. Figures 230b and c illustrate the application of fivefold rotational features into the underlying pentagons and barrel hexagons. This example shares the aesthetics of the more common fourfold swastika designs [Figs. 150a, b and 157a], and examples of this design are found at the Friday Mosque at Yezd (1324) [Photograph 80], in the Topkapi Scroll,Footnote 20 and at the Imam mosque in Isfahan (1611-38).

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Fig. 229

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Fig. 230

Figure 231 illustrates modifications to the standard two-point pattern created from the same rhombic underlying tessellation. Figure 231a shows the relationship between the underlying tessellation and the standard two-point design, and Fig. 231b shows a widened interweaving line treatment of this pattern. In this illustration, the widths of the interweaving lines are maximized so that the acute angles of the superimposed kite elements touch one another. Figure 231c arbitrarily introduces the star rosette into alternating underlying decagonal regions, whereas Fig. 231d applies the geometric rosette to each decagonal region. Figure 231d shows the most frequently encountered version of the classic fivefold design, and an early example is found in the tympanum of an arch over the entry of the Gunbad-i Alayvian in HamadanFootnote 21 (1150-1200) [Photograph 22]. Later Mamluk examples include the Sultan Qaytbay Sabil-Kuttab in Cairo (1479), and the Qadi Abu Bakr Muzhir complex in Cairo (1479-80). An especially attractive Mamluk use of this design is from a stone mosaic panel in the Metropolitan Museum of Art in New York CityFootnote 22 [Photograph 47]. Figures 231e and f represent a modification that changes the angle of declination of the pattern lines within the underlying decagons to be less acute than the standard design in Figure 231a. This design modification was open to stylistic variation [Fig. 225d], and the particular proportions of this illustration were used by artists during the late Abbasid period at the Mustansiriya in Baghdad (1227-34).

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Fig. 231

3.1.7 Fivefold System: Wide Rhombic Repeat Unit

Patterns created from the fivefold system make use of several varieties of repetitive stratagem. As with the previous examples of the standard fivefold designs in each of the four pattern families, the most common repeat unit is the wide rhombus with 72° and 108° included angles. The most basic underlying tessellation that employs this repeat unit is comprised of just decagons and concave hexagons. Figure 232 illustrates the designs produced from this underlying tessellation in each of the four pattern families. Figure 232a shows an acute pattern that is not historical. The two small elements within the concave hexagon are out of balance with the rest of the design. The widened line version in Fig. 232b, while interesting, does not comport with the aesthetics of the fivefold system. Figure 232c employs the 72° angular openings of the median family. However, the visually dominant features of the resulting design in Fig. 232d are the 108° angles of the pentagons and concave octagonal shield shapes that are characteristic of the obtuse family. Indeed, Fig. 232e demonstrates how this same design—the classic obtuse pattern—is created from the underlying tessellation of decagons, pentagons, and barrel hexagons as per Fig. 229a. As demonstrated earlier, the well-known acute pattern in Fig. 232g can be produced with either 108° angular openings, as per Fig. 232f, or with 36° angular openings in the alternative underlying tessellation in Fig. 232h. Interestingly, and as demonstrated, this can also be created as an arbitrary modification of the standard median pattern [Fig. 223]. The derivation in Fig. 232f requires an additive geometric rosette, whereas that of Fig. 232h produces this feature automatically from each midpoint of the alternative underlying polygonal edges. (Historical examples of this design are provided in the text associated with Fig. 227f.) The classic two-point pattern in Fig. 232j can also be made from the underlying tessellation of just decagons and concave hexagons, although the pattern lines have more contact points with the underlying tessellation in Fig. 232k. (Historical examples of this design are provided in the text associated with Fig. 231a.)

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Fig. 232

The five designs in Fig. 233 repeat upon the same rhombic grid. As illustrated, the two upper patterns can be created from either of two reciprocal underlying polygonal tessellations, and follow the dual qualities demonstrated in Fig. 200. The underlying tessellation in Fig. 233a is comprised of decagons, long hexagons, and concave hexagons, and the applied pattern lines have 72° angular openings placed at the midpoints of each polygonal edge. This angular opening is ordinarily associated with the median pattern family. However, the visual character of the resulting design in Fig. 233b is identifiably obtuse, as per the 108° angular opening of the pattern lines applied to the underlying decagons, pentagons, barrel hexagons, and thin rhombi in Fig. 233c. This is a very-well-known design, and two relatively early examples include a late Abbasid carved stucco arch spandrel at the Palace of the Qal’a in Baghdad (c. 1220), and a Seljuk Sultanate of Rum courtyard portal at the Agzikara Han in Turkey (1242-43). Mamluk examples include the stone mosaic in the mihrab niche of the Sultan Qala’un funerary complex in Cairo (1284-85), and a frontispiece from the Quran commissioned by Sultan Faraj ibn BarquqFootnote 23 (1399-1411) [Photograph 48]. A Muzaffarid cut-tile mosaic example is found in the main portico of the Friday Mosque at Kerman (1349). These same two reciprocal underlying tessellations can also be used to create the two-point pattern in Fig. 233e. The angles of declination of the pattern lines located at each underlying polygonal edge in Fig. 233d have 72° while those of Fig. 233f have 36°. This two-point design was used by Mamluk artists in the mihrab at the Sultan Qansuh al-Ghuri complex in Cairo (1503-05). As demonstrated in Fig. 197, the arrangement of six underlying pentagons surrounding the thin rhombi in Fig. 233c and f, while well suited to both the obtuse and two-point families, do not generate acceptable pattern features within the acute and median families. Figure 233g illustrates an acute pattern (by author) created from the modified underlying tessellation that replaces four of the six pentagons with trapezoids, thereby creating a large central wide rhombus that is contiguous with the long edges of the trapezoids. The resulting acute pattern does not appear to have been used historically, which is surprising in that this method of modifying the underlying tessellation was relatively well known [e.g. Fig. 245d], and this rather basic example results in a very attractive pattern. Figure 233h shows an acute design that employs the modification to the underlying tessellation that replaces all six of the pentagons that surround the thin rhombi with trapezoids, leaving a large concave hexagon as an interstice region at the center of the repeat. As mentioned previously, this well-known pattern can also be created from the underlying tessellation of just edge-to-edge decagons and concave hexagons shown in Fig. 232f, as well as through the modification of median pattern lines as per Fig. 223f. (Historical examples of this fine design are provided in the text associated with Fig. 227e.) The median pattern in Fig. 233i is also created from the same underlying tessellation as the acute pattern in Fig. 233h. This design employs two points upon each of the long edges of the underlying tessellation, and while a median pattern, it has distinctive characteristics of the two-point family. This fine example was used by Timurid artists at the Ghiyathiyya madrasa in Khargird, Iran (1438-44).

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Fig. 233

Figure 234 illustrates an obtuse design from the Seljuk Sultanate of Rum that is found at the Gök madrasa in Tokat (1275-80). This design repeats on a rhombic grid and can also be created from two different underlying tessellations. Figure 234a shows the underlying tessellation comprised of decagons surrounded by a ring of ten pentagons, with thin rhombi and large irregular decagonal interstice regions. The 108° angular opening of the pattern lines at each midpoint of the polygonal edge creates the pentagons contained within underlying pentagons that characterize the obtuse family. Figure 234c demonstrates how this same obtuse design can be produced from an alternative tessellation comprised of decagons, long hexagons, half-concave hexagons, and interstice regions with applied 72° angular openings. The visual appeal of Fig. 234b is augmented by the influence of the interstice regions upon the completed design. Figure 235 illustrates two versions of another obtuse design that repeats upon a rhombic grid and can be created from either of the two underlying tessellations. Figure 235c shows a Mamluk interlocking tiled version from a Quranic frontispiece (1313) illuminated by Aydoğdu bin Abdullah al-Badri and Ali bin Muhammad al-Rassam.Footnote 24 This is a relatively rare example of a geometric design produced by known artists. Figure 235d shows a Seljuk Sultanate of Rum interweaving version from the Huand Hatun mosque complex in Kayseri (1237) in which the pattern lines have been widened to the maximum amount, allowing the lines to touch corner to corner with other widened lines. Figure 235a demonstrates the relationship between this pattern to the underlying generative tessellation of decagons, pentagons, barrel hexagons, and thin rhombi, whereas Fig. 235b shows the same design created from underlying decagons, long hexagons, and concave hexagons. Figure 236 shows a median pattern that repeats upon the same rhombic grid and is created from an atypical underlying tessellation comprised of decagons, pentagons, wide rhombi, and triangular elements that are half of a wide rhombus. Unusually, the edge length of the decagons is equal to the newly created long edge of the triangular modules. Typically, the edges of the underlying pentagons that surround the decagon are coincident with the decagonal edges. In this case the decagons and pentagons meet at their vertices. The experimental spirit exhibited in this pattern provided the fivefold system, and indeed other design systems, with increased originality and diversity. This design was used by Kartid artists at the Shamsiya madrasa in Yazd (1329-30), as well as by Timurid artists at the Ulugh Beg madrasa in Samarkand (1417-20). The median pattern in Fig. 237 also orientates the underlying pentagons such that their vertices face the centers of the ten-pointed stars. The underlying tessellation of this design is comprised of pentagons, long hexagons, wide rhombi, atypical octagons, and atypical ten-pointed star polygons. The underlying octagons (pink) in this figure can be substituted with a combination of two wide rhombi, four half-concave hexagons, and a central thin rhombi as per the underlying tessellation in Fig. 209a. The 10-pointed star interstice region at each corner of the repeat unit is essentially the same as the half wide rhombi and decagons from Fig. 236, except for the treatment of the applied pattern lines. This design is from the Seljuk Sultanate of Rum and is found at the Sultan Han in Kayseri (1232-36) [Photograph 42].

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Fig. 234

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Fig. 235

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Fig. 236

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Fig. 237

There are two historical designs created from the fivefold system that repeat upon the rhombic grid with 72° and 108° angles that have particularly large amounts of geometric information within their respective repeats. The underlying tessellations of both of these examples share the same basic structure,Footnote 25 but differ in the application of secondary underlying polygonal modules, and infill treatments of the underlying decagons. Both of these examples are the work of Seljuk artists: one from the Seljuk Sultanate of Rum, and the other from the Great Seljuks in Persia; and both are masterpieces of geometric design. The obtuse pattern in Fig. 238 is from the cut-tile mosaic ornament of the Karatay madrasa in Konya (1251-52). Figure 238a illustrates the concept behind the underlying generative tessellation. This is based upon the well-known rhombic tessellation of decagons separated by two edge-to-edge pentagons, with barrel hexagons completing the coverage (black) [Fig. 200c]. Edge-to-edge underlying decagons are placed at each vertex of this initial polygonal structure. This produces a distinctive ring of ten decagons at the vertices of the repetitive rhombic grid. Figure 238b applies further polygonal infill into the central regions of the initial pentagons and barrel hexagons, as well as into arbitrarily selected secondary decagons. It is worth noting that the large interstice region at each repetitive vertex that was not provided with further polygonal infill was a purposeful exclusion. Figure 238c applies the pattern lines into the underlying tessellation in Fig. 238b. There are two additive features to this pattern line application: the ten-pointed stars and rosettes at the vertices of the repetitive grid, and the underlying decagons that remained unfilled in Fig. 238b. The central motive of the former introduces the rosette and ten darts in the same basic formula as demonstrated in Fig. 222. The treatment of the pattern lines within the open underlying decagons follows the Seljuk practice detailed in Fig. 224a. The end result in Fig. 238d is an exceptionally well-worked-out geometric design of considerable complexity. Figure 239 illustrates the repetitive layout and underlying generative tessellation of the remarkable raised brick design on the façade of the Gunbad-i Qabud in Maragha (1196-97) [Photograph 24]. This tomb tower is decagonal in plan, and the geometric pattern created from this underlying tessellation is applied continuously across nine of the ten sides: the tenth side being the portico. What is more, in a visual tour de force, and feat of artistic dexterity, the pattern flows uninterrupted across the ten engaged columns at each corner of the tomb tower. Like the example from the Karatay madrasa in Konya, Fig. 239a shows how the starting point of this design is the identical placement of edge-to-edge decagons at each vertex of the standard tessellation of decagons, pentagons, and barrel hexagons. Figure 239b fills in this decagonal network with long hexagons, concave hexagons, and wide rhombi from the fivefold system. Three of the secondary decagons located on the vertices of the primary barrel hexagon are not provided with secondary infill, nor are the decagons that are placed upon the vertices of the rhombic repetitive grid at the base of the tessellation in Fig. 239c. Keeping the three decagons unfilled provides this tessellation with reflection symmetry along the long vertical axis of each rhombic repeat unit, but not along the shorter horizontal axis. This type of break in symmetry is very unusual within this tradition. The highlighted lower region in Fig. 239c represents the portion of this tessellation that was used in Maragha. It is noteworthy that the artist who devised this design chose to cut the pattern off above the horizontal line of symmetry. This is another unusual form of symmetry break. Figure 239d illustrates the underlying tessellation in four of the nine uninterrupted linear repetitive cells that span nine of the ten sides of this tomb tower. Whereas the artist kept the set of decagons that rest upon the repetitive rhombic vertices along the lower edge of the design, the similarly placed decagons in the upper set of repetitive vertices have been filled in with long hexagons and concave hexagons. This is yet another break in symmetry. With this further infill of these decagons, the rhombic repeat in Fig. 239c no longer has translation symmetry, and to achieve translation symmetry with the tessellation in Fig. 239d one would have to mirror the tessellation upon the upper horizontal line of symmetry—thus achieving a rectangular repeat unit with clear dodecagons at each corner. Figure 239e shows the underlying tessellation throughout all nine of the ten sides that received this pattern. The grey zones represent the regions of the design that wrap around the ten engaged columns at each corner of the tower. These regions are half circles in plan. Figure 240a illustrates the median pattern as applied to 4/9 of the underlying generative tessellation of the design from the Gunbad-i Qabud. Each of the unfilled underlying decagons from Fig. 239d has been provided with the relatively common Seljuk fivefold rotational motif [Fig. 224a]. This arbitrary additive modification affectively transforms this to a field pattern, and the fact that the fivefold symmetry of these modifications does not align with the vertical reflective symmetry of the multiple 1/9 divisions is another break in symmetry. In fact, this entire geometric construction can be thought of as an exercise in symmetry breaking. Figure 240b illustrates the same 4/9 segment of this design without the underlying generative tessellation. It is worth noting that the design from the Gunbad-i Qabud includes an additive secondary level of pattern in the background of the primary design [Fig. 67] [Photograph 24]. This provides a further level of complexity and visual interest, and is an early outlier of the dual-level design aesthetic that developed in the same general region during the fifteenth century.

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Fig. 238

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Fig. 239

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Fig. 240

3.1.8 Fivefold System: Thin Rhombic Repeat Unit

Although less common, the diverse types of repeat units employed within the fivefold system also include the rhombus with 36° and 144° included angles [Fig. 5b]. Figure 241b illustrates an obtuse pattern that was used by artists working during the Seljuk Sultanate of Rum at the Muzaffar Buruciya madrasa in Sivas (1271-72), as well as by Mamluk artists at the Amir Altinbugha al-Maridani mosque in Cairo (1337-39). Figure 241a shows how this design can be produced from an underlying tessellation of decagons, pentagons, barrel hexagons, and thin rhombi. This derivation employs the 108° angular opening of the crossing pattern lines associated with the obtuse family. Figure 241c demonstrates how this same design can be created from a tessellation of decagons, long hexagons, and concave hexagons, with 72° angular openings. Figure 242 illustrates an acute design with atypical irregular pentagons incorporated into the underlying generative tessellation. The earliest known use of these unusual underlying pentagons, and the distinctive pattern motif they create within the acute family, is from one of the recessed arches in the upper portion of the northeast dome chamber in the Friday Mosque at Isfahan (1088-89) [Fig. 261] [Photograph 25]. The two nonconforming edge lengths of these pentagons have a φ proportional relationship to the standard edge length, and share the same longer edge length as the adjacent scaled-up wide rhombi. The extension of the acute pattern lines from the irregular pentagons into the adjacent wide rhombi creates crossing pattern lines with 72° angular openings. This change from 36° angular openings to 72° angular openings constitutes a regional change from the acute family to the median family. Transitions between pattern families within a single design are unusual, and invariably rely on manipulations in the scale of the underlying polygonal modules [Figs. 269 and 270]. This design is relatively well known, with early examples being produced contemporaneously by artists during the Seljuk Sultanate of Rum at the Huand Hatun madrasa in Kayseri (1237) and by Kartid artist at the Turbat-i Shaykh Ahmad-i Jam in Torbat-i Jam, northeastern Iran (1236). Later Mamluk examples include the minbar of the khanqah and mosque of Sultan al-Ashraf Barsbay (1432-33), and a minbar door in the collection of the Victoria and Albert Museum in London.Footnote 26 The 36° acute angles of this repeat unit allow the rhombus to become a 1/10 segment of a decagon when cut in half between the two vertices at the obtuse included angles. This 1/10 triangle can then be provided with tenfold rotation symmetry to create a decagonal radial design. The pierced marble radial design on the sides of the Ottoman minbar at the Selimiya complex in Edirne, Turkey (1568-74), employs a design that rotates and copies the half repeat unit in Fig. 242 in just such fashion. The patterns in Fig. 243 employ an underlying tessellation with contiguous truncated decagons placed upon each vertex of the rhombic repeat unit. This arrangement can also be achieved through overlapping the decagons [Fig. 191b]. This creates a series of continuous linear bands comprised of truncated decagons that run through the vertices of this rhombic grid. The design demonstrated in Fig. 243a is a two-point design that is unusual in that it utilizes double sets of 72° crossing pattern lines placed at two locations on each of the edges of the underlying pentagons. This provides for 54° angles of declination rather than either 72° or 36° that are far more common to two-point patterns created from the fivefold system. This feature creates the large and small five-pointed stars that are characteristic of the median family at each of the underlying pentagons. The precise placement of the 72° crossing pattern lines on the underlying pentagonal edges is so contrived as to allow for the selected extended lines of the smaller five-pointed stars to intersect with the midpoints of the long edges of the truncated decagons. This in turn allows for the creation of the ten-pointed star rosette at the center of each truncated decagon. Figure 243b demonstrates the aesthetic success of this highly unusual and ingenious design. This pattern was used as a border in the late Abbasid main entry portal of the Mustansiriyah madrasa in Baghdad (1227-34). It is worth noting that the generative schema of a much later Ottoman design from a door of the Sultan Bayezid II Kulliyesi in Istanbul (1501-06) is remarkably similar to this design from Baghdad [Fig. 270]. Figure 243d illustrates a Mamluk two-point design that is produced from the same underlying tessellation as the two-point design from the Mustansiriyah madrasa in Baghdad. Figure 243c demonstrates the application of the standard two-point pattern lines with 36° angles of declination. This two-point design is from the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81). Figure 244 illustrates a more complex Mamluk two-point pattern that also repeats upon the same acute rhombic grid. This exceptional example is from a minbar (1468-96) commissioned by Sultan Qaytbay that is in the collection of the Victoria and Albert museum in London. This design is created from an underlying tessellation of decagons, pentagons, barrel hexagons, and thin rhombi. Star rosettes have been arbitrarily added into the ten-pointed stars located at the vertices of the rhombic repeat.

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3.1.9 Fivefold System: Rectangular Repeat Units

Many patterns created from the fivefold system repeat upon a rectangular grid. Figure 245 features six such designs. Figure 245a shows an obtuse pattern that can be easily created from the rectangular tessellation of decagons, pentagons, barrel hexagons, and thin rhombi. The first known use of this popular pattern is the work of Qarakhanid artists, and is found in the rear portico of the Maghak-i Attari mosque in Bukhara (1178-79). Later examples include the Sultan Han in Aksary, Turkey (1229); the Friday Mosque at Ashtarjan, Iran (1315-16); the Shah-i Zinda funerary complex in Samarkand, Uzbekistan (1386); and the Abdulla Ansari complex in Gazargah, Afghanistan (1425-27). Figure 245b shows a two-point pattern from the Mughal mausoleum of Humayun in Delhi (1562-72) [Photograph 79]. Like the illustration, this Mughal example keeps the regions within the ten-pointed stars open. By contrast, the two-point pattern in Fig. 245c incorporates the arbitrary design modification that places ten-pointed star rosettes within the central underlying decagonal regions. This particular variety of star rosette modification is relatively uncommon [Fig. 225c]. This variation is a Mamluk design from the mihrab of the Qadi Abu Bakr Muzhir complex in Cairo (1479-80). As with other designs created from the fivefold system, each of the designs in Fig. 245a–c can also be created from an alternative underlying tessellation, in this case comprised of decagons, long hexagons, and concave hexagons (not shown). As shown, the underlying thin rhombi surrounded by six pentagons in Fig. 245a–c work exceedingly well with the obtuse and two-point pattern families. However, this arrangement of underlying polygons does not work nicely with acute and median patterns. The acute pattern in Fig. 245d is created from an underlying tessellation that replaces the thin rhombus and four of the pentagons with four trapezoids and a central large wide rhombus. This design is from a zillij panel at the Bu ‘Inaniyya madrasa in Fez (1350-55). The acute design (by author) in Fig. 245e replaces the underlying rhombus and all six pentagons with six trapezoids and a large concave hexagon. While not known to the historical record, this is very similar to several historical examples that make use of this alteration of the underlying tessellation, for example, that of Fig. 233h. Figure 245f shows a median design produced from the same underlying tessellation as in Fig. 245e. This is also an ahistorical design (by author) that is fully compatible with the aesthetics of this ornamental tradition. These two types of alteration to the underlying tessellation are demonstrated in Fig. 198. As with many obtuse designs created from the fivefold system, the obtuse design in Fig. 246 can be created from two different underlying tessellations. Figure 246a employs an underlying tessellation comprised of decagons, pentagons, barrel hexagons, and thin rhombi, while that of Fig. 246c has decagons, long hexagons, and concave hexagons. This is an Ilkhanid pattern from the mausoleum of Uljaytu in Sultaniya, Iran (1307-13). Figure 247b shows a rectangular obtuse pattern produced by artists during the Seljuk Sultanate of Rum. Once again, this design can be created from two different underlying tessellations. Figure 247a employs an underlying tessellation comprised of decagons, pentagons, barrel hexagons, and thin rhombi, while that of Fig. 247c has decagons, long hexagons, and concave hexagons. This pattern is found at the Sirçali madrasa in Konya (1242-45), as well as at the Ahi Serafettin mosque in Ankara (1289-90). The two-point pattern in Fig. 248b was used on three roughly contemporaneous Mamluk wooden minbars: one commissioned by Sultan Qaytbey (1468-96) and currently in the collection of the Victoria and Albert Museum in London; the minbar of the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81); and the minbar of the Amir Azbak al-Yusufi complex in Cairo (1494-95) [Photograph 46]. This two-point design can be created from either of the two underlying tessellations. Figure 248a employs an underlying tessellation comprised of decagons, pentagons, barrel hexagons, and thin rhombi, while that of Fig. 248c has decagons, long hexagons, and concave hexagons. As with other examples with reciprocal underlying tessellations, these are essentially duals of one another. The pattern within the underlying decagonal regions of the rectangular repeat employs ten-pointed star rosettes [Fig. 225b]. The obtuse pattern in Fig. 249b has an unusually large amount of geometric information within each rectangular repeat unit. This was created during the Seljuk Sultanate of Rum for the Yusuf ben Yakub madrasa in Cay, Turkey (1278). Figure 249a shows the construction from an underlying tessellation comprised of decagons, pentagons, barrel hexagons, and thin rhombi, while Fig. 249c employs decagons, long hexagons, concave hexagons, conjoined long hexagons, and half-concave hexagons. The pattern lines at the center of the conjoined long hexagons do not rest conveniently upon the edges of this underlying polygon. However, the pattern lines in this same region of the underlying tessellation in Fig. 249a sit precisely upon the midpoints of the underlying polygonal edges. It therefore appears more likely that the artist responsible for this design employed the construction illustrated in Fig. 249a. The acute design in Fig. 250b is from a Mamluk metal window grille at the al-Azhar mosque in Cairo. This design is created from an underlying tessellation of decagons, pentagons, barrel hexagons, and long hexagons. This arrangement places decagons at each vertex of the rectangular repeat, as well as at the center of each repeat. As mentioned previously, not all polygonal modules within the fivefold system work nicely in each of the four pattern families. As highlighted previously, a case in point is the long hexagon within the acute family [Fig. 187]. The artist responsible for this design sought to ameliorate the constrained conditions of the acute pattern lines within the long hexagon by adjusting the angles of the pattern lines as they enter the long hexagon, resulting in these pattern lines being noncollinear. This is an example of the willful departure from convention to achieve a more pleasing design. However, a general rule of this ornamental tradition is for crossing pattern lines to remain collinear at the point were they intersect with one another. While at first glance this appears as an acceptable modification, upon closer inspection, the noncollinearity is somewhat problematic. The two-point pattern in Fig. 250d is created from the same underlying tessellation. In Fig. 250c, four of the parallel lines within the long hexagons are much closer together when compared with the pattern density of the rest of the design, and this would appear to be a problem. However, the Mamluk artist who produced this two-point pattern widened the line in only one direction: outward from the adjacent parallel line, thereby avoiding the problem of pattern density. The highlighted region in Fig. 250d represents a carved stone relief panel from the Sultan Qaytbay Sabil in Jerusalem (1482). This relief panel further reduces any remaining sense of constraint within the parallel pattern lines of the long hexagon by only using a quarter of this module within the finish relief panel. Figure 250e illustrates another use of this same underlying tessellation by Mamluk artists. The resulting obtuse pattern in Fig. 250f was used on the railing of the stone minbar of the Sultan Barquq mausoleum in Cairo (1384-86) [Photograph 57]. Figure 251a illustrates the derivation of a rectangular median pattern created from an underlying tessellation comprised of decagons, pentagons, barrel hexagons, long hexagons, concave hexagons, and wide rhombi. This combination of underlying polygons is somewhat unusual in that it combines the underlying long hexagons and concave hexagons with underlying pentagons and barrel hexagons. Within the fivefold system, the 72° crossing pattern lines associated with the pentagons and barrel hexagons create distinctive features of the median pattern family, particularly the five-pointed stars. However, within the long hexagon and concave hexagon the 72° crossing pattern lines produce features that are characteristic of the obtuse family, such as the pentagons, concave decagons, and shields associated with these modules. The combination of these elements within a single underlying tessellation therefore produces features that are both median and obtuse. Figure 251b shows an interweaving version of this design that was produced by artists during the Seljuk Sultanate of Rum for the Külük mosque in Kayseri (1280-90). Figure 252b illustrates a Mamluk two-point pattern from the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81). Figure 252a demonstrates how this pattern is created from an underlying tessellation that places decagons at the vertices and center of the rectangular repeat unit, and pentagons, barrel hexagons, and thin rhombi in the connective polygonal field. The pattern lines in Fig. 252b have been widened to their maximum outward expansion so that the outer corners of the independent kites, rhombi, and concave hexagons meet at a single point. Figure 253 illustrates a Timurid median pattern from the Imam Reza complex in Mashhad, Iran (1405-18). The underlying generative tessellation in Fig. 253a is unusual in that it has two sizes of decagon. The edge length of the smaller decagons is standard to this system by virtue if its edge lengths equaling those of the barrel hexagons and wide rhombi. The long edge of the triangular element determines the size of the larger decagons, and these atypical modules are simply wide rhombi that have been divided into half. This larger variety of underlying decagon is also seen in the earlier Kartid example from Yazd [Fig. 236]. Figure 254b shows a Timurid acute design from the Shah-i Zinda complex in Samarkand (fourteenth century). Figure 254a shows how the cluster of underlying trapezoids and triangles produces the distinctive eight-sided shield motif located at the midpoint of the longer side of the rectangular repeat unit. This arrangement of trapezoids and triangles is especially relevant to the acute pattern family [Fig. 196]. Figure 254d illustrates another pattern that includes this distinctive motif, albeit in a much more complex geometric structure. Figure 254c shows how the same arrangement of underlying polygons responsible for this feature is similarly placed at the midpoints of the long edge of the rectangular repeat unit. This design includes features that are derived from two configurations of contiguous triangles: one associated with 2/10 of the decagon, and the other with 6/10 [Fig. 196]. This example is considerably more complex and was produced by Muzaffarid artists for the Friday Mosque at Kerman (1349). Figure 255b shows an acute pattern that was used by Mamluk artists on the minbar (1300) of the al-Salih Tala’i mosque in Cairo, as well as by Mughal artists at the mausoleum of Akbar in Sikandra, India (c. 1612). Figure 255a shows how this design is derived from an underlying tessellation that places decagons at the vertices and center of the rectangular repeat, with connective pentagons, barrel hexagons, trapezoids, and triangles from the fivefold system. The repetitive structure of this design is such that the geometric information contained within the rectangular repeat unit is identical to that of its dual. Like the previous example, this design also employs groupings of two edge-to-edge underlying triangles that combine to equal a 2/10 portion of the decagon. Figure 256b shows a Shaybanid acute design from a door of the Kukeldash madrasa (1568-69) in the Lab-i Hauz complex in Bukhara [Photograph 78]. A later Janid Khanate example was used at the Bala Hauz mosque in Bukhara (1712). Figure 256a shows how this is created from an underlying tessellation comprised of decagons, pentagons, barrel hexagons, trapezoids, and triangles. Each of the long edges of the trapezoids is contiguous with a long edge of a triangle. This example includes distinctive pattern motifs that are created from the underlying partial decagons made up of 2/10 and 3/10 groupings of the triangular modules [Fig. 196].

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3.1.10 Fivefold System: Hexagonal Repeat Units

Most patterns created from the fivefold system that repeat upon a hexagonal grid are field patterns [Figs. 212–220]. However, patterns with ten-pointed stars can also use hexagonal repeat units, although very few are known to the historical record. Figure 257a illustrates a particularly successful example of a median design that employs a hexagonal repeat unit that places the underlying decagons upon the midpoints of the long edges of the repeat unit. This is unusual in that the primary underlying polygonal modules are ordinarily located at the vertices of the repetitive grid. The distribution of ten-pointed stars is the same as patterns that repeat on a rhombic grid with 36° and 144° included angles [Figs. 241–244], and technically this design can be said to repeat on this same rhombic grid. However, for the purposes of this study and for reasons of clarity, repeat units are identified as having reflected symmetry in the pattern lines located along their edges. The tenfold radial symmetry of the ten-pointed stars in this design does not align with the neighboring ten-pointed stars that are separated with an underlying wide rhombus module. Rather, the pattern lines between these ten-pointed stars are askew from one another and lack reflected symmetry. Again, for the purposes of this study, their lack of reflected symmetry therefore precludes this design from being categorized as repeating on a rhombic grid. This skewed feature between the ten-pointed stars is relatively uncommon and creates a pleasing visual tension within the design. It is worth noting that this interesting design can also repeat with a rectangular cell (not shown) that contains the area of two hexagonal repetitive cells. While the design in Fig. 257b is very successful, the artist from the Seljuk Sultanate of Rum who created this pattern arbitrarily filled each ten-pointed star with the relatively common modification that introduces five-pointed stars [Fig. 224b]. This results in the very balanced field pattern in Fig. 257c from the Huand Hatun in Kayseri (1237). Figure 258b shows a Mamluk two-point pattern that repeats upon a hexagonal grid. The underlying tessellation shown in Fig. 258a places the decagons at each vertex of the repeat unit, with edge-to-edge decagons at the two short edges of the repeat, and decagons separated by two mirrored pentagons along the other four longer edges. The other underlying polygonal modules within this generative tessellation are pentagons, wide rhombi, and an unusual stellated interstice region at the center of the repeat unit. The small highlighted rectangular region in Fig. 258b represents the isolated region of this design that was used in the entry portal of the Ashrafiyya madrasa in Jerusalem (1482). By only using this limited region of the overall design, the artist successfully circumvented the problem of the tightly constrained parallel pattern lines that result from the application of two-point pattern lines to the underlying wide rhombi [Fig. 187]. This rather clever solution is conceptually identical to that employed in the stone relief panel from the Sultan Qaytbay Sabil in Jerusalem [Fig. 250d]. The fact that both these idiosyncratic Mamluk examples are from Jerusalem and were produced in the same year strongly indicates the likelihood of their being produced by the same artist or atelier. Figure 259b illustrates an unusual Ilkhanid design from the Gunbad-i Gaffariyya in Maragha, Iran (1328). Rather than placing the pattern lines upon either the midpoints or two points of the underlying polygonal edges, this design utilizes the vertices of the underlying tessellation. As such, it does not conform to any of the four pattern families. The underlying generative tessellation is comprised of a linear band of truncated decagons and pentagons, and is identical to the underlying tessellations of the two designs in Fig. 243. Whereas the earlier examples repeat with the thin rhombic repeat associated with this system, the atypical application of pattern lines in this example are more closely associated with the hexagonal dual of the rhombic repeat. The linear band of partial decagons at the top and bottom of the panel have a different set of applied pattern lines than the central set of overlapping decagons, and this breaks the translation symmetry of the hexagonal grid. To add to the eccentricity of this design, the widened line treatment combines interweaving and interlocking qualities.

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3.1.11 Fivefold System: Radial Designs

Of all the historical design systems, the fivefold system was the most widely used for creating patterns with rotation symmetry. This variety of design was afforded four distinct manners of application. The first is the most obvious: the production of standalone panels that are usually circular, and typically have a ten-pointed star at the center. Figure 260b shows just such a design, produced during the Khwarizmshahid period and located at the Zuzan madrasa in northeastern Iran (1219) [Photograph 40]. As illustrated and like so many designs created from the fivefold system, this example can be easily created from either of two underlying tessellations. A better known example of a stand-alone tenfold radial design is the aforementioned pierced marble circular panel on the sides of the Ottoman minbar at the Selimiya complex in Edirne, Turkey (1568-74). Each 1/10 radial segment of this panel is half of the rhombic repeat unit that was first used in several locations by artists during the Seljuk Sultanate of Rum [Fig. 242]. Radial designs with five- or tenfold symmetry were occasionally used on the flat soffits of muqarnas constructions. This of course requires the shape of the soffit to also have compatible rotation symmetry: for example, pentagons, decagons, five-pointed stars, or ten-pointed stars. The most geometrically interesting examples of fivefold radial pattern making are in the secondary infill of the primary design elements in dual-level designs. Again, the primary elements receiving the secondary infill will themselves have fivefold rotation symmetry, including pentagons, decagons, five-pointed stars, and ten-pointed stars. Marinid examples of such fivefold rotational elements within dual-level designs are found at both the Bu’Inaniyya madrasa (1350-55) and the al-‘Attarin madrasa in Fez, Morocco (1323) [Figs. 474 and 476]. Timurid, Qara Qoyunlu, and Safavid artists created several dual-level designs that contain regions with either fivefold or tenfold rotation symmetry, including examples from the Imamzada Darb-i Imam in Isfahan (1453) [Fig. 451] [Photograph 97], and the Madar-i Shah in Isfahan (1706-1714) [Figs. 453 and 468]. The single most significant group of fivefold dual-level designs with regions of rotation symmetry are the five examples from the Topkapi Scroll.Footnote 27 Two fine examples from the Topkapi Scroll include a ten-pointed star [Fig. 22a] that is part of a dual-level design comprised of an obtuse primary pattern and a median secondary patternFootnote 28, and a second example that is self-similar in that both the primary and secondary patterns are from the same median family [Fig. 449]. The last variety of radial pattern created from the fivefold system involved the application of geometric designs onto the surface of domes. Figure 21 illustrates an example from the Samosa Mahal at Fatehpur Sikri, IndiaFootnote 29 (seventeenth century) that used 8/10 of a two-dimensional radial design to create a conical form that was then used on the eight gore segments of a dome. A small amount of distortion invariably results from this method of applying geometric designs to the surfaces of domes. The Topkapi Scroll depicts several gore segments that are presumably intended for application to domes,Footnote 30 including the design created from the fivefold system represented in Fig. 260e. The underlying generative tessellation for this acute design is comprised of decagons, pentagons, barrel hexagons, and clustered triangles.

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3.1.12 Fivefold System: Hybrid Designs

As with the fourfold system A and fourfold system B, the fivefold system was also used to create hybrid designs that contain more than a single repetitive cell within their overall composition. Hybrid designs bring greater complexity to a given design, and their composition requires greater geometric aptitude. Examples of hybrid designs generally date to the period when this ornamental tradition was reaching full maturity during the thirteenth century, with notable twelfth-century exceptions such as the fourfold system A design at the Maghak-i Attari mosque in Bukhara (1178-79) [Fig. 155], and the fourfold system B design at the Mu’mine Khatun in Nakhichevan, Azerbaijan (1186) [Fig. 182]. However, it is remarkable that the earliest known hybrid design predates these examples by a century. Among the different patterns that were included in the upper recessed arches in the northeast dome chamber of the Friday Mosque at Isfahan (1088-89) is the fivefold hybrid design in Fig. 261b that is composed of at least two separate repetitive cells [Photograph 25]. This is all the more remarkable in that along with two other examples from the northeast dome chamber [Figs. 229a and 496] these are the earliest extant fivefold Islamic geometric designs known to the historical record. This hybrid design has a central pentagonal region with fivefold rotational symmetry that is attached to rhombic cells on either side of the pentagon. These rhombi have 72° and 108° included angles, and the pattern lines associated with each is the classic acute pattern [Fig. 226b]. Because the boundary of the arch limits the amount of geometric design, only portions of the rhombic cells are contained within the arch. Due to this limitation, the structure of the repetitive cells is somewhat ambiguous and must be inferred—especially where the pattern moves outward from the central arched region. What is certain is the use of the pentagonal and rhombic cells within the arch itself. The 36° leftover space between the rhombic cell at the apex of the arch is a 1/10 segment of the circle, and easily filled with an added point of the ten-pointed star, or conceivably with a second rhombus with 36° and 144° included angles (as shown). The underlying tessellation is demonstrated in Fig. 261a. This employs five irregular pentagons at the center of the pentagonal repetitive cell. This produces the distinctive arrangement of pattern lines wherein a central pentagon is surrounded by 5 nine-sided motifs derived from the five-pointed star. This pattern line configuration shares the properties of a popular design from the Huand Hatun madrasa in Kayseri, Turkey (1237) [Fig. 242b]. Figure 262d illustrates a Seljuk Sultanate of Rum hybrid design from the Huand Hatun in Kayseri. Figure 262a illustrates the two repetitive cells that combine to create this design: the rhombus and hexagon (dashed lines). Either of these will produce very good patterns when used on its own. In fact, the pattern within just the rhombus is the very-well-known classic obtuse design. It would appear significant that the Huand Hatun also contains a remarkable hybrid pattern created from the fourfold system A that also has just two repetitive elements [Fig. 156]. Without question, the artists responsible for the geometric ornament of this building were exceptionally skilled, with a sophisticated understanding of hybrid design methodology. Figure 262b represents an interweaving treatment of the standard median pattern created from the underlying tessellation in Fig. 262a.Footnote 31 Figure 262b shows how this hybrid design repeats upon a rectangular grid that places ten-pointed stars at the vertices and 2 ten-pointed stars within the field of each repeat unit (dashed lines). The pattern in Fig. 262d includes an arbitrary modification of the standard design that places an additive fivefold motif within each of the ten-pointed stars [Fig. 224a], affectively transforming the design with ten-pointed stars into a field pattern. As indicated in Fig. 262c and d, the artist’s decisions regarding the rotational orientation of the fivefold additive modifications within the ten-pointed stars changes the otherwise rectangular repeat to one that is twice as long as the unmodified design (dashed lines). In addition to the decagons, the underlying polygonal modules within the hexagonal repetitive elements include wide rhombi and half-concave hexagons that are clustered around small thin rhombi. While rare, this configuration of half-concave hexagons and small thin rhombi was used in other locations during the Seljuk Sultanate of Rum [Figs. 209, 215, 217 and 218]. Figure 263c shows a more complex fivefold hybrid obtuse design from the Izzeddin Keykavus hospital and mausoleum in Sivas (1217). Figure 263a shows the four repetitive cells that make up this design. These include a large rhombus with 72° and 108° included angles, a smaller rhombus with the same proportion, a rhombus with 36° and 144° included angles with the same edge length as the larger wider rhombus, and a triangle that is half the thinner rhombus. The short edge of the triangle is the same length as the edge of the smaller wide rhombus. The small wide rhombus is identical to that of Fig. 262a and on its own produces the classic fivefold obtuse design. As with all repetitive cells used within this tradition, it is required that the underlying polygonal structure and the resulting applied pattern lines of each repetitive edge must match all other edges of equal length. In this case, there are two edge lengths. The short lengths have underlying edge-to-edge decagons placed on each vertex, and the longer edge lengths are populated with two underlying decagons separated by a concave hexagon placed in its long orientation. It is worth noting that the underlying polygonal modules that generate this design could also be the dual of this tessellation, comprised of decagons, pentagons, barrel hexagons, and thin rhombi [Fig. 200]. Figure 263b illustrates the hybrid tessellation of these four repetitive elements, and Fig. 263c shows how this combination has translation symmetry with a rectangular repeat. Figure 264c shows 1 of 2 fivefold hybrid designs from the Karatay Han near Kayseri (1235-41). As shown in Fig. 264a, this acute pattern is comprised of four repetitive cells: the barrel hexagon, thin rhombus, wide rhombus, and a triangle that is half a wide rhombus. There are two edge lengths in this group, each with its own underlying polygonal configuration. On its own, the wide rhombus makes the classic acute pattern. Figure 264b indicates an interstice region within with underlying tessellation of the repetitive barrel hexagon. This creates an interesting, if not completely successful, feature in the derived pattern. Had the artist employed the 3/10 cluster of three triangles [Fig. 196], this would be a more satisfactory design. However, the use of clustered triangles, each being 1/10 of a decagon, to create visually appealing acute patterns did not come into use until the fourteenth century, and for all their geometric skill and artistry, it appears that this seemingly simple innovation was not known to the thirteenth century artists working on this building. Figure 264c shows how the arrangement of the four repetitive elements combine together in a rectangular repeat unit with translation symmetry. Figure 265c shows a second hybrid acute design from the Karatay Han near Kayseri that was doubtless created by the same artist as the previous example. Figure 265a illustrates the six repetitive cells that comprise this design. The first four of these are identical to those of Fig. 264. There are three edge lengths in this group, each with its own underlying polygonal configuration. The barrel hexagon repetitive cell employs the same atypical, and not wholly satisfactory, interstice region as the design from Fig. 264. The two underlying long hexagons (green) in the second repetitive hexagonal cell produce pattern lines that are not generally acceptable within the acute family [Fig. 187]. The 36° crossing pattern lines produce constrained regions within the applied pattern lines associated with this particular hexagon. This is one of the only historical examples of an acute design that employs this unsatisfactory motif. The panel in Fig. 265c represents the full rectangular repeat unit that this particular combination of repetitive elements creates. Figure 266 illustrates a Mughal hybrid pattern from a stone mosaic panel on the façade of the I’timad al-Daula in Agra, India (1622-28). This type of pattern was not typical to the Mughal period, but it is nonetheless very successful. The three repetitive cells are simply the rhombus with 72° and 108° included angles, a triangle that is half of the rhombus, and a rectangle with a short edge length that is equal to the long edge of the triangle. Once again, the pattern lines within the rhombi are the classic acute design. The pattern lines within the rectangle provide this example with its most distinctive feature: the rotational point symmetry at the center of the rectangular cell. This is created from an arrangement of 12 edge-to-edge underlying triangles that form an s-curve with point symmetry at its center. The acute pattern in Fig. 267b was produced by Mamluk artists for the Qadi Abu Bakr ibn Muzhir in Cairo (1479-80). This exceptional fivefold design is one of the very few Islamic geometric patterns to incorporate 20-pointed stars into a matrix of 10-pointed stars. The overall repeat unit with translation symmetry is the wide rhombus with 72° and 108° included angles and 20-pointed stars at each vertex (dashed line). However, Fig. 267a demonstrates how the repetitive structure of this design is comprised of an arrangement of repetitive decagonal and concave hexagonal cells. Some of the vertices of these decagons and concave hexagons have partial underlying decagons made up of six adjacent triangles that equal 6/10 of a decagon [Fig. 196]. The center of each decagonal repetitive cell houses a 20-pointed star. The incorporation of 20-pointed stars within the fivefold system is analogous to the incorporation of 16-pointed stars within the fourfold system A and fourfold system B [Figs. 165–168, and 185–186]. Figure 268b illustrates a Marinid acute hybrid pattern with 10- and 20-pointed stars that is conceptually similar to the previous Mamluk example. This outstanding design is from the Bu ‘Inaniyya madrasa in Fez, Morocco (1350-55). Figure 268a shows how the hybrid repetitive cells of this example are made up of decagons and rhombi with 72° and 108° included angles. Figure 268b indicates the rectangular repeat unit with translation symmetry. The rectangular dual of this repeat unit (not shown) has the identical geometric information as the repeat unit itself, which is to say that the pattern is self-dualing. As with the Mamluk example, the 10-pointed stars are located at the vertices of the hybrid structure, and the 20-pointed stars are placed at the center of each repetitive decagonal cell. Once again, on its own, the rhombic repetitive cell is the repeat unit for the classic fivefold acute pattern [Fig. 226b]. This design from the Bu ‘Inaniyya madrasa is arguably one of the most beautiful fivefold acute patterns from this ornamental tradition.

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3.1.13 Fivefold System: Patterns with Variable Scale

A very unusual and very rare use of the fivefold system derives the pattern lines from two scales of underlying polygonal modules within a single underlying tessellation. The design in Fig. 269b is from a door produced during the Seljuk Sultanate of Rum for the Hekim Bey mosque in Konya (1270-80), and currently in the Ince Minare Medrese History Museum in Konya. Figure 269a shows how the underlying tessellation transitions between polygonal modules of two different scales. The smaller scale polygons, as represented by the smaller decagons and surrounding pentagons, have pattern lines from the acute family. As these acute pattern lines extend into the adjacent larger scale wide rhombi, decagons, long hexagons, and concave hexagons, the pattern lines naturally convert to the median family. The proportional relationship between these two scales of underlying polygon is 1.3764…. This is a product of φ: the inherent proportional relationships within the fivefold system. If the small decagonal and pentagonal edges are taken as 1, the length of a line that connects any two nonadjacent corners of the pentagon is 1.6180… [Fig. 195]; and if the short diagonal of a rhombus with 72° and 108° included angles is 1.6180… then the edge of this rhombus is 1.3764…. The pattern lines within the larger underlying decagonal modules are provided with an arbitrary modification that disguises the ten-pointed star in a similar fashion as in Fig. 224b. The introduced five-pointed stars at the center of each large underlying decagon are identical in size and shape to the five-pointed stars associated with the smaller scaled pentagons. This provides visual similitude between the acute region and the obtuse region, and helps to unify the design. Filling the large ten-pointed stars with this motif also provides the design with a more balanced density throughout. This design is one of the few historical examples of this variety of pattern manipulation, which is surprising in that the incorporation of variable scaled pattern elements offers tremendous innovative appeal. However, the aesthetics of this ornamental tradition generally seeks to achieve an overall balance in design density, and the fact that regions of reduced scale in the underlying generative tessellation will typically cause concomitant regions of greater density within the resulting pattern matrix would likely have been an aesthetic impediment to the development of this category of design. Traditional aesthetics aside, it is possible for diminishing scale within a geometric design to be visually appealing and intellectually satisfyingFootnote 32 [Figs. 484 and 485], albeit very deferent from historical examples. Figure 270c shows a design from the Sultan Bayezid II Kulliyesi in Istanbul (1501-06) that is also created from an underlying tessellation with variable scaled polygonal modules, and also transitions between two different pattern families. Figure 270b demonstrates how the smaller scale polygons have pattern lines from the acute family, while the larger scale polygons are populated with the two-point family, with two sets of crossing pattern lines with 72° angular openings. This creates the less acute five-pointed stars with points of 72° that are more commonly associated with the median family, thereby introducing the characteristics of a third pattern family into this one design. Figure 270a demonstrates how the proportional relationship between both scales of the underlying polygonal modules is derived from a ring of ten smaller pentagons fitting precisely within the larger scaled decagon. The black lines in Fig. 270a illustrate the truncation lines that create the trapezoids in Fig. 270b. This allows for the dart-shaped acute pattern lines within the smaller scaled trapezoids to extend into the larger scaled pentagons, thereby determining the placement of the two points upon each edge for the larger scale two-point pattern lines. The proportion of the two edges lengths is 1.9021…. As with the example from the Hekim Bey mosque, this is a product of φ. If the sides of the small underlying decagons and pentagons are taken as 1, then the distance between two outer points in the smaller ring of ten edge-to-edge pentagons that surround the decagon is 1.9021, the length of the large pentagonal edges. This also equals the distance between two consecutive corners of the small underlying decagon. The scale of the larger underlying polygons is almost double that of the smaller. By doubling the amount of pattern lines application to the larger underlying polygons through the use of two-point application with lines that continue in both directions beyond the underlying polygonal edges, the artist who designed this outstanding pattern successfully balanced the overall design density within both regions of variable scale. Both in terms of its visual splendor and geometric ingenuity, as well as for the excellent quality of the woodworking, this example of Ottoman geometric ornament is nothing short of a masterpiece.

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3.1.14 Sevenfold System

Patterns created from the sevenfold system are characterized by the presence of heptagons, 7-pointed stars, and 14-pointed stars. These are rare, with every historical example known to the author being included within this study.Footnote 33 As with the other historical design systems, the sevenfold system employs a limited set of underlying polygonal modules to which pattern lines are applied in each of the four pattern families. Considering their beauty, the rarity of this variety of design would not appear to be due to any aesthetic predilection against their appearance. Rather, one must conclude that knowledge of this system was held and passed on to only a very few select artists within those Muslim cultures that included such designs within their ornamental canon. Considering their broad spread over time and territory, it appears likely that the discovery of the sevenfold system may have occurred independently in several locations rather than as a continuum of inherited knowledge. Considering the paucity of historical examples, it is impossible to know for certain to what extent artists working with sevenfold patterns were aware of this as a design system per se, or were merely applying the methodology of the polygonal technique to sevenfold geometry to arrive at stand-alone patterns without realizing the systematic potential of the underlying polygons within the generative tessellations they produced. This historical ambiguity in no way diminishes the fact that each of the historical examples can be created from the limited set of underlying generative polygons that comprise the sevenfold system, nor the fact that this system has extraordinary potential for creating countless original designs for contemporary artists.

As with the fivefold system, the earliest example of a pattern created from underlying polygonal modules of the sevenfold system is from the Seljuk work on the northeast dome chamber of the Friday Mosque at Isfahan (1088-89). Soon after this example, two Ghaznavid sevenfold patterns were incorporated into the exterior façade of the minaret of Mas’ud III in Gazna, Afghanistan (1099-1115). Following this, there is a hiatus of roughly a century before several rather simple sevenfold patterns were produced in Anatolia during the Seljuk Sultanate of Rum. It is possible that all three of these locations may have been isolated developments of sevenfold pattern making. The added sophistication of incorporated 14-pointed stars did not transpire until another century had passed. Mamluk artists appear to have independently developed the more fully expanded set of underlying polygonal modules that comprise this system in the early fourteenth century. Indeed, the majority of historical examples of sevenfold patterns with 14-pointed stars are Mamluk, with a few notable Ottoman and Timurid examples that were presumably influenced by their earlier Mamluk precursors.

Figure 271 shows the five types of underlying generative polygons that comprise the sevenfold system. These consist of the heptagon and tetradecagon, the two regular polygons native to this system; those that result as interstice regions through tessellating with other polygonal modules from this system; those that result from truncating the heptagon or tetradecagon; those that result from the intersection of the heptagon or tetradecagon; and those that result from the union of the heptagon or tetradecagon. The examples illustrated in this figure are not exhaustive, and there are many more interstice and truncation modules than shown. It is also worth noting that not all of the polygonal modules shown were used historically, and that the sevenfold system offers tremendous innovative opportunity to contemporary artists seeking to create original geometric designs.

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Fig. 271

Figure 272 illustrates the pattern line applications to the two regular polygons of the sevenfold system. The angular openings of the pattern lines are determined by drawing lines that connect the midpoint of the sides of the tetradecagon, ranging from adjacent sides [14-s1] through six sequential sides [14-s6].Footnote 34 The two-point pattern lines are shown as a 14-s4 edge-to-edge sequence, but other two-point sequences are also possible. With the six possible midpoint-to-midpoint line sequences [14-s1–14-s6], the acute, median, and obtuse pattern family assignments are less specific than with the other pattern systems. As such, the designation of the pattern family in the sevenfold system is generally descriptive of the aesthetic character of a given pattern rather than the specific angular opening employed. The line sequence nomenclature [14-s1–14-s6] is, therefore, necessary for accurately identifying the precise character of any given pattern created from this system. The median, obtuse, and two-point tetradecagons are provided with additive geometric rosettes that are typical to this ornamental tradition. However, other varieties of rosette, and other treatments of the 14-pointed stars are also possible, and these are only meant as representative examples. Figure 273 shows the interstice polygonal modules with associated pattern lines in each of the four pattern families. Again, this is only a representative sample of the interstice modules that are generated from this system. Unlike Fig. 272, only one midpoint-to-midpoint line sequence is shown for each family. These are the more commonly employed within the limited number of historical designs generated from this system. Figure 274 demonstrates the pattern lines applied onto the truncated polygonal modules. Again, this is only a representative sample of the truncation modules that are generated from this system, and only one of the midpoint-to-midpoint line sequences is shown for each family. Truncated tetradecagons were not a feature of historical methodological practices. However, when used in rotation with matching truncated edges, they can provide a positive design contribution.Footnote 35 Note: those truncated tetradecagons that have no applied pattern lines do not make acceptable design features within the 14-s2 obtuse and 14-s4 two-point families. Figure 275 shows the polygonal modules that are derived from intersections of the heptagon and tetradecagon. Only the more visually acceptable midpoint-to-midpoint line sequences are shown for each family. The two modules without pattern lines do not generate acceptable features within the obtuse family. Figure 276 illustrates the polygonal modules that are derived from the union of both heptagons and tetradecagons along with their associated pattern lines in each of the four pattern families. Several of the conjoined tetradecagons have no pattern lines as these polygonal modules do not work well with the particular variety of line sequence.

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The geometric properties of the sevenfold system are governed by the inherent proportional ratios of the heptagon. Figure 277 demonstrates how the heptagonal edge, taken as 1, relates to ρ (rho) as the length that connects two consecutive edges (1.80193774…), and σ (delta) as the length that connects three consecutive edges (2.24697960…). These ratios are analogous to the φ (phi) proportional ratio of the golden section (1.61803398…) that is inherent within the pentagon and which functions analogously as the proportional determinant within the fivefold system. Figure 278 provides several examples of linear arrangements of tetradecagons and various secondary modules of the sevenfold system. This demonstrates how the sevenfold proportions of the heptagon determine the tessellating properties created from the polygonal modules of this generative system. Each interval of two tetradecagons (either overlapping, edge to edge, or separated by secondary polygonal modules) can be used on its own as an edge configuration for both rhombic and rectangular repeat units. The proportions of the linear arrangements of three tetradecagons as shown in this figure are especially relevant to the design of the smaller scaled secondary pattern when creating dual-level designs from the sevenfold system.

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Figure 279b illustrates an acute field pattern that is one of the motifs in the group of arches in the upper portion of the square base of the northeast dome in the Friday Mosque at Isfahan [Photograph 26]. The only other known example of this design is from the anonymous treatise On Similar and Complementary Interlocking Figures. This treatise is estimated to date to circa 1300,Footnote 36 and the fact that these are the only known examples of this very distinctive design suggests a direct causal influence between them. As detailed in the previous chapter, it is highly significant that the underlying generative tessellation in Fig. 279a is represented along with the acute pattern in the illustration in this treatise. Considering that this is the earliest known sevenfold pattern, it is somewhat surprising that the polygonal modules that comprise the underlying tessellation do not include the heptagon. Significantly, the absence of the heptagon suggests that the artist who created this design may have understood the systematic potential of the two varieties of underlying hexagon. The underlying hexagons in Fig. 279c are simply created from the intersection of two heptagons. Figure 279d shows how the underlying barrel hexagons can be derived as an interstice region of an arrangement of heptagons and overlapping heptagons. Figure 279e produces the same interstice regions, but with heptagons and hexagons from Fig. 279c; and Fig. 279f produces the barrel hexagon from an interstice region of an arrangement of just the hexagons from Fig. 279c. This last example is the arrangement that was used to produce the design in the northeast dome chamber. These polygonal arrangements demonstrate the determinant nature of the heptagon, and the artist who conceived this design could not have derived the two underlying hexagonal modules without starting with the heptagon. This artist would have therefore been aware of at least three polygonal components, and it is plausible that this artist would have known that these underlying modules could be arranged into other generative tessellations, thereby producing other sevenfold patterns. If this artist was also responsible for the two patterns created from the fivefold system that are in other arches in this same area of the northeast dome chamber—a supposition that would appear most likely—then we can conclude that this artist was knowledgeable of systematic design methodology more generally. It would therefore seem entirely reasonable for this artist to seek a means to also produce sevenfold patterns systematically. Figure 279a shows how this design repeats upon either of the two dualing hexagonal repetitive grids. With the lack of any primary star forms, this example falls into the category of field pattern.

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Fig. 279

The exterior façade of the minaret of Mas’ud III in Ghazni, Afghanistan (1099-1115), includes two designs with sevenfold symmetry.Footnote 37 Along with the single example from the northeast dome in the Friday Mosque at Isfahan these are among the earliest examples of complex sevenfold pattern making to have ever been produced.

Figure 280 illustrates the construction sequence for one of the two Ghaznavid sevenfold designs from the façade of the minaret of Mas’ud III in Ghazni, Afghanistan (1099-1115). Figure 280a shows the underlying generative tessellation comprised of edge-to-edge heptagons that repeat upon an elongated hexagonal grid. The interstice regions in this configuration are filled with edge-to-edge irregular pentagons. Figure 280b demonstrates the first step in the placement of the pattern lines. These are unusual in that they are set upon the vertices of the generative grid rather than the midpoints of each heptagonal edge. This pattern was created during the developmental period that preceded the methodological codification of the polygonal technique, and is an excellent example of the artistic experimentation prevalent under the Ghaznavid patronage. Figure 280c shows the completion of the pattern through the incorporation of secondary pattern lines, and the modification of the primary seven-pointed stars from Fig. 280b to include a seven-pointed star rosette placed at each vertex of the hexagonal repeat. The secondary pattern lines in Fig. 280c elegantly employ 60° angles placed at two points of each polygonal edge. These extend into the underlying pentagons to create the distinctive five-pointed stars. Figure 280d is a representation of the raised brick panel with widened interweaving lines from the façade of this monument. Figure 281 shows the construction sequence for the second sevenfold pattern from the minaret of Mas’ud III in Ghazni. Figure 281a indicates how both the repeat unit and underlying generative tessellation are the same as the previous example from the same building. Figure 281b demonstrates the first step in the placement of the pattern lines. These follow the more conventional approach of placing the crossing pattern lines at the midpoints of the underlying polygons. Figure 281c shows the completion of the pattern through the incorporation of a relatively complex network of secondary pattern lines. The overlapping kite elements provide the aesthetics of the two-point pattern family. Figure 281d shows an approximation of this raised brick design.

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Fig. 280

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Fig. 281

Figure 282 illustrates four patterns created from the same underlying tessellation of edge-to-edge heptagons that was used in the Ghaznavid patterns in Figs. 280 and 281. Each of these four designs utilizes the midpoints of the underlying polygonal edges for pattern line application, and they differ significantly from the two Ghaznavid designs in that the pattern line application is more standardized: wholly determined by the underlying tessellation, without the inclusion of secondary pattern lines. The historical patterns in this figure were produced during the Seljuk Sultanate of Rum roughly a century later than the examples from Ghazni, by which time this ornamental tradition had reached greater maturity and codification. While the pattern line applications are standardized, the simple deployment of edge-to-edge heptagons in the underlying tessellation is not necessarily indicative of a systematic methodology. Certainly, the heptagon and pentagonal interstice elements are both modules from the sevenfold system. However, unlike the two underlying hexagons and implicit heptagon of the example from the northeast dome chamber in Isfahan, the heptagon and interstice pentagons of these four examples will not rearrange into other tessellations on their own. For this reason, despite the fact that these two modules are members of the larger family of polygons contained within the sevenfold system, the artists responsible for the historical designs in this figure were not necessarily aware of the otherwise systematic nature of the sevenfold patterns they constructed. Figure 282a is an acute pattern [14-s5] from the Great Mosque of Dunaysir in Kiziltepe, Turkey (1200-04), as well as at the Alaeddin mosque in Nidge (1223). Figure 282b shows a median pattern [14-s4] that, on its own, is not known within the historical record, but was the basis for the more complex design from the minaret of Mas’ud III in Fig. 281b. Figure 282c shows an obtuse pattern [14-s2] from the Eğirdir Han (1229-36), and Fig. 282d illustrates a two-point pattern from the Great Mosque of Malatya in Turkey (1237-38). These three Anatolian examples were produced within a 38-year period, and it is possible that all three are the work of a single person or artistic lineage.

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Fig. 282

Mamluk artists were the first to develop the sevenfold system into its fully mature expression; with 14-pointed stars, considerable complexity resulting from the large number of underlying polygonal modules, and diverse repetitive stratagems. With the notable exception of the rectangular design at the Sultan al-Mu’ayyad Shaikh complex in Cairo (1412-22), all of the historical designs created from this system during its mature expression repeat upon rhombic grids. There are three rhombi that are the product of sevenfold symmetry [Fig. 10]. Figure 283 shows grids made up of these three rhombi along with applied 14-s4 median patterns (by author). Only the medium rhombus with 2/14 and 5/14 included angles and the wide rhombus with 3/14 and 4/14 included angles were used historically as repeat units. Unlike the fourfold and fivefold systems, none of the known historical sevenfold designs made use of hybrid repetitive cells. However, the methodology of the polygonal technique is also well suited to providing greater design diversity through hybrid repetitive constructions with the sevenfold system. This is especially relevant to contemporary artists with an interest in expanding the repertoire of the sevenfold system. To this end, the patterns in each of these three rhombic repeat units in Fig. 283 have identical edge configurations, allowing them to also be used in combination with one another. The design in Fig. 284 employs all three of the rhombi from Fig. 283. This hybrid 14-s4 median design (by author) has overall hexagonal translation symmetry (dashed lines). The design in Fig. 285 uses the same three rhombi in an array that tessellates the tetradecagon with 14-fold radial symmetry. It is interesting to note that these three rhombi can also be used to cover the plane non-periodically.

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Fig. 285

The earliest known example of a design created from the more mature expression of the sevenfold system is from one of the stone lintels in the south elevation of the Qawtawiyya madrasa in Tripoli, LebanonFootnote 38 (1316-26). This design, along with its underlying generative tessellation is illustrated in Fig. 286a. This is a 14-s2 obtuse pattern in which the underlying tetradecagons are filled with a star rosette that follows the convention of the fivefold system [Fig. 222]. Figure 286b employs the same generative tessellation but with an added ring of 14 trapezoids within the tetradecagons. As with the pentagons in this tessellation, these trapezoids are also truncated heptagons [Fig. 271]. This is a 14-s4 median pattern that was used in a carved stucco panel at the Amir Burunduq mausoleum at the Shah-i Zinda complex in Samarkand (1390-1420), as well as in a carved stone panel in the exterior façade of the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81). Figure 286c demonstrates how the same pattern can be created from the dual of the tessellation in Fig. 286b, in which case this design can be categorized as a 14-s4 acute pattern. In both cases, the polygonal modules of their respective tessellations are members of the sevenfold system. Both of the designs in Fig. 286 repeat upon the medium rhombic grid with 2/14 and 5/14 included angles [Fig. 10b]. Figure 287 illustrates the only other historical design known to repeat upon this rhombic grid. Figure 287a shows the 14-s2 obtuse pattern along with the underlying generative tessellation comprised of tetradecagons, concave hexagons, and edge-to-edge triangles. Figure 287b shows a widened line version of this pattern that was used in a side panel of the wooden minbar in the Sultan Barsbay complex at the northern cemetery in CairoFootnote 39 (1432). Figure 287c shows a subtractive variation of this design that removes the pattern lines from one of the underlying triangular modules. This earlier version was used on an Ottoman wooden door of the Bayezid Pasa mosque in Amasya, Turkey (1414-19).

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Fig. 286

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Fig. 287

Following its rise to full maturity, the majority of designs created from the sevenfold system repeat upon a rhombic grid with 3/14 and 4/14 included angles [Fig 10a]. The design with prominent 14-pointed stars separated by twin 5-pointed stars in Fig. 288 repeats upon this grid. This is from a Mamluk door at the Sultan Qansuh al-Ghuri complex in Cairo, Egypt (1503-05) [Photograph 49]. Figure 288a illustrates the derivation of this pattern from an underlying tessellation of edge-to-edge tetradecagons and interstice concave decagons. The pattern lines associated with this tessellation are categorized as 14-s1 of the obtuse family with an additive 14-fold rosette within each tetradecagon. Figure 288b employs an alternative generative tessellation of tetradecagons, pentagons, trapezoids, and concave hexagons; and the associated pattern lines in this derivation fall into the 14-s6 acute family. As is often the case, these alternative tessellations have a dual relationship. The additive process for creating the central 14-pointed stars in Fig. 288a is very straightforward, and one has to assume would have been well within the skill set of any artist work at this level of sophistication, and the original designer of this very successful design is as likely to have used one of these underlying tessellations as the other. Figure 289c illustrates an acute design from the ceiling of the courtyard cistern at the Suleymaniya mosque in Istanbul (1550-58) [Photograph 81]. This same basic design was used in the courtyard of a house that belonged to a Christian trader in Aleppo (1757), albeit with a different arbitrary treatment applied to the centers of the 14-pointed stars. Figure 289a illustrates how this pattern repeats upon the rhombic grid with 3/14 and 4/14 included angles, and is generated from an underlying tessellation comprised of tetradecagons placed at each repetitive vertex separated by a barrel hexagon on each repetitive edge, trapezoids and central concave decagonal interstice regions. Of course the barrel hexagons are the equivalent of two contiguous trapezoids, and the outer long edges of this arrangement of trapezoids are identical to that of Fig. 288a. Figure 289b shows how the vertices of the two kite motifs located within the interstice regions are fixed upon the vertices of the underlying trapezoids. This is an unusual feature, but highly effective in these circumstances. Figure 289c includes the design modifications at the center of the 14-pointed stars that are present in the example from the Suleymaniya. This involves extending every other line of the central 14-pointed star so that a 7-pointed star is created, and further filling each 7-pointed star with a ring of pentagons surrounding a central heptagon. These seven pentagons actually have a functional purpose: they enclose nodules that spray water from the ceiling into the pool of water within the cistern. The underlying generative tessellation in Fig. 290a is identical to that of Fig. 289a except that the interstice regions have been filled with two varieties of triangle and a central rectangle. Other than this region with introduced triangles and rectangle, the applied pattern lines in Fig. 290b are identical to the design in Fig. 289b, and remain in the 14-s6 acute family. The pattern lines associated with the central rectangle produce an octagon that is almost regular. This design is from the minbar doors of the Haram al-Ibrahimi in Hebron, Palestine. This minbar was produced by Fatimid artists for the Mashhad Nabi Hussein in al-Majdal Asqalan, PalestineFootnote 40 (1191-92), and moved to its current location in Hebron by Ṣalāḥ ad-Dīn a century later. However, the pattern with 14-pointed stars on the minbar doors is distinctly Mamluk, and clearly a later addition, as is the pattern with 12-pointed stars on the back panel of the minbar’s platform. Figure 290c employs and interweaving line as per the historical example from Hebron. The underlying tessellation in Fig. 291a provides an alternative polygonal infill of the concave decagonal interstice regions from Fig. 289a. This infill is comprised of four kite-shaped quadrilaterals and an irregular hexagon at the center. This design was recorded by Bourgoin,Footnote 41 but its location is unattributed. Considering that Bourgoin was working in Egypt, Syria, and the Levant, this design is presumably Mamluk, but could also be Ottoman. Figure 291b demonstrates the placement of the pattern lines onto the underlying polygonal edges, and other than the region of the underlying kites and hexagon, this pattern is the same as Figs. 289 and 290. As such, it is also in the 14-s6 acute family. Figure 291c represents the completed design with interweaving lines as per Mamluk and Ottoman convention. The two designs in Fig. 292 are produced from the same underlying tessellation of tetradecagons separated by concave hexagons, with conjoined heptagons in the center of each repeat. The repeat unit for this design is the same rhombus with 3/14 and 4/14 included angles. Figure 292a illustrates a 14-s2 obtuse pattern from the minbar door at the ‘Abd al-Ghani al-Fakhri mosque in Cairo (1418). The underlying conjoined heptagons are responsible for the two point-to-point seven-pointed stars that are a distinctive feature of this design. This design was also collected by Bourgoin,Footnote 42 but its location is similarly unattributed; and Bourgoin’s work was the likely source for Ernest Hanbury Hankin’s analysis that includes the underlying generative tessellation, but remains unattributed.Footnote 43 Figure 292b shows a 14-s4 2-point design that was used on the wooden congregational Quran stand in the Sultan Qansuh al-Ghuri complex in Cairo, Egypt (1503-05). This is the only known two-point pattern created from the sevenfold system during the period of full maturity. The design in Fig. 293 is 1 of only 2 known sevenfold examples that originate in the eastern regions during the period of full maturity, the other being the example in Fig. 286b. This also utilizes the rhombic repeat unit with 3/14 and 4/14 included angles, and is found in the Timurid shrine complex of Imam Reza in Mashhad, Iran (1405-18). Figure 293a shows the underlying generative tessellation comprised of tetradecagons separated by long hexagons, with pentagons and shorter hexagons within the center of each rhombic repeat unit. Figure 293b shows the derivation of the 14-s3 median pattern. This is readily apparent as analogous to the median designs created from the fivefold system. Figure 293c shows the completed pattern with interweaving lines as per the historical example.

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Fig. 288

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Fig. 289

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Fig. 293

The design in Figure 294 is the only known historical sevenfold example that repeats upon a rectangular grid. The proportions of each rectangular repeat unit appear at first glance to be a square, but are actually not quite equilateral. This beautiful design is from the side panels of the minbar in the Sultan al-Mu’ayyad Shaykh complex in Cairo (1412-22) [Photograph 50]. Figure 294a illustrates the unusual non-aligned edges of the adjacent tetradecagons where they are separated by two mirrored triangles. Figure 294b demonstrates how the pattern lines in this region have an interesting skewed dynamic that results from this nonaligned triangular configuration. This is a 14-s2 obtuse pattern with a 14-fold star rosette in the center of each tetradecagon. Figure 294c represents the interweaving design as per the Mamluk historical example.

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Fig. 294

3.2 Nonsystematic Patterns

As mentioned previously, patterns created from the polygonal technique fall into two broad categories: those whose underlying tessellations are systematic, and those that are nonsystematic. As expounded in the previous section, systematic design methodology utilizes a limited set of polygonal modules that are assembled into diverse tessellations onto which associated pattern lines are placed. By contrast, nonsystematic patterns are derived from underlying tessellations comprised of polygons that are specific to the tessellation and will not reassemble into additional arrangements. Both types of underlying tessellation allow for the creation of designs in each of the four pattern families. Nonsystematic designs vary in complexity between those with only a single variety of primary star form, to those that include multiple regions of local symmetry expressed as multiple star forms within a single pattern. Indeed, it was through the nonsystematic use of the polygonal technique that Muslim artists were able to produce the extraordinary patterns with such unusual combinations as 7- and 9-pointed stars, 9- and 11-pointed stars, 11- and 13-pointed stars, and sequential combinations such as 9-, 10-, 11-, and 12-pointed stars. Nonsystematic design methodology exploits the full range of repetitive strategies, including orthogonal and isometric grids, rhombic grids, rectangular grids, and both regular and irregular hexagonal grids. Without wanting to diminish the remarkable achievements in the historical use of the five design systems, the creative energy spent on developing the nonsystematic design methodology resulted in many of the most geometrically sophisticated and innovative patterns known to this ornamental tradition.

Nonsystematic patterns frequently include the more common primary stars with 6, 8, 10, and 12 points. These are also standard features among systematic design methodology: 6- and 12-pointed stars to the system of regular polygons, 8-pointed stars to both the fourfold system A and the fourfold system B, and 10-pointed stars to the fivefold system. Patterns with less accessible star forms, for example, those with 9, 11, 13, and 15 points are invariably nonsystematic. In examining diverse design methodologies, the previous chapter details why the polygonal technique is the only traditional methodology that allows for the creation of patterns with these more enigmatic symmetries. More specifically, the previous chapter examines the question of whether Muslim geometric artists were dependant upon mathematicians for instructions in the use of conic sections and/or approximate constructions for accurately producing the higher order regular polygons that enable the creation of these types of stars. Historians of Islamic art and mathematics have tended to overlook a very basic, yet very accurate method of creating these otherwise problematic polygons. Figure 295 demonstrates a simple, fast, and effective method of drawing polygons that otherwise requires complex mathematical procedures to construct. This method approximates the regular polygon through dividing the circumference of a circle into the requisite number of segments with a compass or pair of dividers. Step 1 of Fig. 295a demonstrates the drawing of the heptagon by dividing a 1/8 segment of a circle (45°) into eight approximately equal parts (halves, quarters, eighths). The compass is then set at a length that is approximately 1/8 larger than the 45° segment. Step 2 places marks progressively around the circle seven times. The last mark will have a slight shortfall of the vertical starting point. Step 3 divides the shortfall into seven approximate equal parts so that the compass setting is increased by the 1/7 division. This is then used to re-remark the circle from the initial vertical position. Step 4 creates the heptagon by connecting the new marked divisions. Figure 295b demonstrates the drawing of the nonagon in the same fashion, except that the first setting of the compass is decreased by 1/8 rather than increased. If, during Steps 2 and 3, the compass falls beyond the vertical starting point, decrease the compass setting by an amount equal to a division of the long-fall by the number of sides of the intended polygon. This same approximate method can be used with equal ease to draw higher order polygons such as those with 11, 13, 14, 15, etc. sides. This approximate technique is as accurate from a practical standpoint as using conic sections. By way of example, if one is making an enneagon with conic sections, while the end result will be theoretically precise, the actual drawing will only be as accurate as one’s drawing skills and equipment allow. People are not computers, and such drawings will inevitably have inaccuracies, and these inaccuracies will be no less that those resulting from the approximate technique outlined above. With practice, in two or three incremental steps, one can quickly divide the circle into less tangible divisions that are, for all intents and purposes, functionally accurate.

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Fig. 295

Nonsystematic design methodology has three sequential phases: the construction of a radii matrix; the making of the underlying generative tessellation; and the extraction of the geometric pattern. Radii matrices provide a very effective means of constructing the underlying generative tessellations of both simple and complex designs, and evidence from the Topkapi Scroll indicates that these were used to set up the underlying tessellations for both nonsystematicFootnote 44 and systematic designs.Footnote 45 The radii matrices present in the Topkapi Scroll are invariably un-inked lines (dead lines) scribed into the surface of the paper with a steel stylus. In the case of systematic designs, as explained previously, the underlying tessellations are comprised of modular polygonal elements, with associated pattern lines, that are assembled into different combinations. However, in laying out a given combination for placement of a design on a wall or on paper (such as a Quranic frontispiece or the Topkapi Scroll), radii matrices provide a very effective means of accurately drawing the underlying tessellation, whether it be systematic or nonsystematic.

The radii matrix establishes the regions of local symmetry within a given design. For nonsystematic patterns of low complexity this might be the 12-pointed stars on the vertices of either an isometric or orthogonal repetitive grid. With more complex patterns, these regions of local symmetry allow for the placement of different stars with n-fold symmetry at the repetitive vertices, centers of the repeat, midpoints of the repetitive edges, and/or within the field of the repeat unit. Figure 296 demonstrates the use of a radii matrix to construct the well-known fivefold underlying tessellation that repeats upon a rhombic grid. The construction of tessellations from radii matrices typically begins with the establishment of the pentagons, followed by the primary polygons with n-fold local symmetry (in this case decagons), followed by interstice regions (in this case the barrel hexagon). The radii matrix in Fig. 296 is associated with the fivefold system. Although this is systematic, the proportional regularity inherent within the fivefold system provides a useful demonstration of the ideal relationship between the radii matrix and its resulting tessellation. In this example, the pentagons are regular, and the edge lengths of all the polygonal elements are the same. When this methodology is applied to nonsystematic pattern generation the pentagonal proportions and edge lengths invariably become irregular. However, as a general rule, the closer they are to the ideal—as exemplified by the characteristics of fivefold symmetry—the better the quality of the geometric pattern that is produced from the generative tessellation. Conversely, the greater the disproportion within the generative polygons the less likelihood of creating a successful pattern. It is important to note that in nonsystematic pattern making, while the pentagons, hexagons, and other elements within the polygonal matrix that separate the primary polygons are not regular, the primary polygons with n-fold local symmetry are always regular. This provides for the regularity of the n-pointed stars that characterize this design tradition. Step 1 of Fig. 296 shows an array of 20 radii placed at each vertex of the standard wide rhombus associated with fivefold symmetry, with the lines extended into the rhombus until they meet with other extended lines. This methodology is often assisted by placing twice the number of radii (in this case 20) as there are number of sides to the primary polygon (in this case 10). In Step 2 a circle is drawn that is tangent to the red radii, and lines are drawn that intersect the circle and are perpendicular to the two blue radii that meet at the center of the circle. Step 3 shows the regular pentagon that these two lines create, as well as the two decagons created by rotating each line ten times around their respective rhombic vertex. Step 4 mirrors the pentagons and decagons, thereby creating the barrel hexagon at the center of the rhombus. Step 5 shows the completed underlying tessellation without the radii. Note: The edges of the pentagons are congruent with the red radii.

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Fig. 296

Just as a single underlying tessellation will create multiple geometric patterns, one of the potent features of radii matrix design methodology is the ability of a single radii matrix to generate more than one underlying polygonal tessellation. This provides for a relatively large number of geometric patterns that can be created from a single radii matrix. Figure 297 shows the construction of another well-known fivefold underlying tessellation that is created from the same radii matrix as that of Fig. 296. The difference between the constructions of these two underlying tessellations is in the position of the pentagons relative to the 20 radii at each rhombic vertex. In this example, the pentagons have congruent edges with the blue radii rather than the red radii. In each case, the first objective is to create the pentagons, and from the pentagons, the decagons. Step 1 shows how this is achieved by drawing the lines that connect the vertices of the red and blue radii, and mirroring these on the indicated red radii (dashed lines). Step 2 demonstrates how the pentagons are established by following the same method as the previous example: by introducing circles that are tangent to the relevant radii and applying lines that are perpendicular to the radii. Step 3 shows how the pentagonal edges create the decagons, and Step 4 produces the thin rhombus, and barrel hexagons on each repetitive edge through mirroring the pentagons and decagons. Two features of this tessellation are the primary polygons (in this case decagons) being separated along the edge of the repeat unit by barrel hexagons, and the cluster of six pentagons surrounding the thin rhombus. Each of these common fivefold features is also encountered frequently in the application of this radii matrix based design methodology to nonsystematic pattern generation, although the pentagons will not be regular and the edge lengths of the polygons will not all be identical.

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Fig. 297

The most basic nonsystematic patterns place just one variety of primary star at the vertices of the repetitive grid. The radii matrices for such designs are simple geometric structures, as are the underlying tessellations that they will produce. Figure 298 illustrates an isometric radii matrix that creates underlying tessellations with dodecagons at each vertex of the triangular repeat. Three irregular pentagons clustered at the center of the triangle separate these dodecagons. Step 1 places an array of 24 radii at each corner of the triangle. Step 2 places a circle that is tangent to the three red radii, and places two lines that are perpendicular to the blue radii and intersect the circle and blue radii. Step 3 identifies the irregular pentagon that is implicit in Step 2, and uses the two lines from Step 2 to produce the two dodecagons. Step 4 mirrors these elements to complete the tessellation, and Step 5 illustrates the tessellation without the radii. Figure 299 shows another underlying tessellation created from the same isometric radii matrix as the example in Fig. 298. The dodecagons in this example are separated by barrel hexagons, with a similar cluster of three irregular pentagons at the center of the repeat. As established in Steps 1, 2, and 3, the pentagons in this underlying tessellation are tangent with the blue radii. Figure 300a illustrates the acute, median, obtuse, and two-point patterns created from the underlying tessellation in Fig. 298, while the four patterns in Fig. 300b were created from the underlying tessellation in Fig. 299. The tremendous potency of this methodological practice is demonstrated by the fact that all eight of these designs are created from two underlying tessellations that in turn are produced from just a single radii matrix. Numerous examples of the acute pattern in Fig. 300a are known to the historical record, and locations include the Great Mosque of Niksar in Turkey (1145); the Izzeddin Keykavus hospital and mausoleum in Sivas (1217); the Abbasid Palace of the Qal’a in Baghdad (c. 1220); the Great Mosque of Divrigi (1228-29); the mausoleum of Uljaytu in Sultaniya, Iran (1307-13) [Photograph 82]; the Friday Mosque at Varamin, Iran (1326); the Amir Qawsun mosque in CairoFootnote 46 (1329-1330) [Photograph 52]; the madrasa al-Mirjaniyya in Baghdad (1357); the Taşkın Paşa mosque in Damsa Köy bei Ürgüp, TurkeyFootnote 47 (c. mid-fourteenth century); the Khatuniyya madrasa in Tripoli, Lebanon (1373-74); and the Amir Mahmud al-Ustadar complex in Cairo (1394-95). The median design in Fig. 300a was used at the Mustansiriyah madrasa in Baghdad (1227-34), as well as at the mausoleum of Uljaytu at Sultaniya, Iran (1313-14). The obtuse design in Fig. 300a (by author) is not known to the historical record, but is similar to a well-known and superior pattern created from the system of regular polygons [Fig. 108a]. An example of the two-point pattern in Figure 300a is found at the Ribat Ahmad ibn Sulayman al-Rifa’i in Cairo (1291). The acute design in Fig. 300b was used in multiple locations, including the exterior ornament of the Abbasid Palace of the Qal’a in Baghdad (c. 1220), a pair of Mamluk doors at the al-Azhar mosque, as well as two representations in the Topkapi Scroll.Footnote 48 It is interesting to note that one of the examples from the Abbasid Palace of the Qal’a is a hybrid construction that combines the triangular repetitive cell of this example with a square repeat that shares geometric information along the shared repetitive edges [Figs. 23d–f], and that this same hybrid use of the triangular and square repetitive cells is also shown in diagram number 35 of the Topkapi Scroll. While use of the median pattern in Fig. 300b (by author) is not known historically, this design meets the aesthetic criteria of this tradition. The obtuse pattern in Fig. 300b was used in the Aq Qoyunlu ornament of the Friday Mosque at Isfahan (1475), and the two-point pattern in this figure was used in several Mamluk locations, including a large wooden door in the Sultan al-Mu’ayyad Shaykh complex in Cairo (1415-22); the portal of the Ribat Khawand Zaynab in Cairo (1456); and a carved stone lintel in the Ashrafiyya madrasa in Jerusalem (1482).

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Fig. 298

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Fig. 299

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Fig. 300

Figure 301 demonstrates the construction sequence of four additional underlying tessellations that can also be created from the radii matrix introduced in Fig. 298. The upper two have the same polygonal configuration on each edge of the triangular repeat, whereas the two lower examples have two identical edges and one unique edge. Figure 302 shows the applied pattern lines from each of the four pattern families to the additional underlying tessellations created in Fig. 301. The patterns in the far left column are acute designs, those in the central left column are median designs, those in the central right column are obtuse designs, and those in the far right column are two-point designs. None of these 16 patterns are known to the historical record, and some are more acceptable to traditional aesthetic conventions than others. These 16 designs (by author), in addition to the eight patterns illustrated in Fig. 300, demonstrate the high level of generative potential of just a single radii matrix.

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Fig. 301

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Fig. 302

Figure 303 demonstrates the variety of isometric tessellations that can be created from a single radii matrix comprised of two centers of local symmetry. This radii matrix places 24 radii at each corner of the triangular repeat, and 18 radii at the center of the repeat. Each of the five tessellations places dodecagons at the triangular corners and nonagons at the center of each repeat. These, in turn, produce 12- and 9-pointed stars, respectively. As demonstrated in the previous examples, each of these five tessellations will generate distinct patterns from each of the four pattern families (not shown).

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Fig. 303

Figure 304 demonstrates the variety of orthogonal tessellations that can be created from a single radii matrix comprised of two centers of local symmetry. This radii matrix places 24 radii at each corner of the square repeat, and 16 radii at the center of the repeat. Each of the five tessellations places dodecagons at the corners of the square and octagons at the center of each repeat. These, in turn, produce 12- and 8-pointed stars, respectively. Each of these five tessellations will generate patterns from each of the four pattern families (not shown). Only the tessellations in the top two rows (A and B) are known to have been used historically to generate geometric patterns.

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Fig. 304

Figure 305 demonstrates the variety of rectangular tessellations that can be created from a single radii matrix comprised of two centers of local symmetry. This radii matrix places 24 radii at each corner of the rectangular repeat, and 20 radii at the center of the repeat. Each of the five tessellations places dodecagons at the corners of the rectangle and decagons at the center of each repeat. These, in turn, produce 12- and 10-pointed stars, respectively. Each of these five tessellations will create patterns from each of the four pattern families (not shown). Only the underlying tessellation in the top row is known to have been used historically [Fig. 414].

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Fig. 305

Figure 306 shows three square tessellations that can be created from a single radii matrix comprised of four centers of local symmetry. This radii matrix places 24 radii at the corner of the square repeat, 8 radii at the center of the repeat, 20 radii at the midpoints of the repeat, and 18 radii within the field of the polygonal matrix. These regions of local symmetry correspond to dodecagons, octagons (or regions with fourfold symmetry), decagons, and nonagons respectively. These, in turn, produce complex patterns with 12-, 10-, 9-, and 8-pointed stars (or octagons). As in the previous figures, each of these three tessellations will generate four distinct patterns; one from each of the pattern families (not shown). Only the upper tessellation is known to the historical record [Fig. 400].

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Fig. 306

Figure 307a illustrates the classic acute pattern created from the fivefold system along with a highlighted detail. The five- and ten-pointed stars have five- and tenfold rotational symmetry, respectively, and the applied pattern lines uniformly bisect the midpoints of each underlying polygonal edge. This uniformity is also seen in the standardized 36° angular opening of the crossing pattern lines at each midpoint of the underlying polygonal edge. By contrast, Fig. 307b shows how nonsystematic patterns do not share this inherent uniformity. While both patterns share the same basic configuration of pentagons and barrel hexagons that surround their respective primary polygons, the underlying pentagons in Fig. 307b are not regular, and the polygonal edge lengths are not all equal. As such, the five-pointed stars do not have rotational symmetry. As mentioned previously, when creating a nonsystematic design, the general objective is to create pattern elements that are as close as possible to the ideal proportions exemplified in the fivefold system. In order to achieve this, the placement of the crossing pattern lines upon the polygonal edges of the underlying tessellation will not always be located precisely at the midpoints, but may have to move up or down the polygonal edge in order to produce better looking design proportions. This is demonstrated by the pattern line placements within the green squares in Fig. 307b. Similarly, in order to achieve a more balanced effect, it is also occasionally necessary to move the intersection of the crossing pattern lines slightly off of the underlying polygonal edge, as shown in the crossing pattern lines within the green circles in Fig. 307b. What is more, it is often necessary for the angles of the crossing pattern lines to slightly vary from location to location. The precise placement and angles of the pattern lines are subtle aesthetic decisions made by the artist. The one consistent area of uniformity is in the rotational symmetry of the primary star forms.

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Fig. 307

As nonsystematic patterns become more complex they are more likely to contain design elements that are asymmetrical. Figure 308 is an orthogonal design with 12- and 16-pointed stars, as well as 7-pointed stars within the pattern matrix [Fig. 396b]. Many of the constituent shapes that make up this design are asymmetrical, including the shapes that have been highlighted in blue and green. The aesthetics of this tradition are highly reliant upon symmetry, and patterns with a greater preponderance of asymmetrical components are generally less likely to be pleasing to the eye. However, the visual discord of a given asymmetrical pattern element is rectified through reflection. In this way, asymmetrical elements are paired with identical elements through reflection; thereby providing the symmetry that is fundamental to the visual appeal of this tradition. The blue truncated stars in Fig. 308a demonstrate how a reflected pair can be immediately adjacent to the line of reflection, while the blue truncated stars in Fig. 308b are separated by a similar element that has bilateral reflected symmetry. This is also the case with the green hexagons in Fig. 308a. As a general rule, the closer together the reflected pairs, the more successful the design, but many factors play into the aesthetic success of particularly complex designs, and there are many exceptions to this general rule.

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Fig. 308

3.2.1 Isometric Designs with a Single Region of Local Symmetry

The level of complexity of patterns created from each of the systematic design methodologies is largely a product of the ratio between the number of secondary connective polygons and the single variety of primary polygons that comprise the underlying tessellation. While this general principle is also true of nonsystematic patterns, the number of different primary n-sided polygons within a single underlying generative tessellation is a further variable in determining complexity. In this way, nonsystematic patterns created from an underlying tessellation with primary polygons of a single variety that are connected by a minimal number of secondary polygons will be the least complex, whereas those produced from multiple varieties of primary polygon that are connected by a large number of secondary polygons will be the most complex.

Complexity and beauty should not be conflated. The three patterns in Fig. 309 are not particularly complex, but each is well balanced and imbued with visual interest. Each is created from the same nonsystematic underlying tessellation comprised of two edge-to-edge regular pentagons placed at the midpoints of each edge of the triangular repetitive cell such that the outer corners of the twinned pentagons touch. This creates two interstice regions: one that is a six-pointed star located at the vertices of the isometric grid, and the other a shield-shaped ditrigon located at the center of each triangle. The angular openings of the applied pattern lines in Fig. 309a are determined in part by the strategic placement of regular hexagons centered on each vertex of the isometric grid. Each edge of these hexagons contributes to the formation of regular heptagons placed at the outer points of each underlying six-pointed star. This Seljuk design, with its distinctive heptagons, is from the upper arches in the base of the northeast dome chamber in the Friday Mosque at Isfahan (1088-89) [Photograph 27]. Figure 309b shows more or less the same design, but with slightly different angular openings within the applied pattern lines, and without the hexagon centered on the vertices of the isometric grid. The lack of these hexagons disallows the heptagonal motif, and transforms the ring of 6 five-pointed stars from the previous pattern to the ring of 6 ten-sided elongated motifs that surround the central six-pointed star. This design is from the anonymous treatise On Similar and Complementary Interlocking Figures in the Bibliothèque Nationale de France in Paris.Footnote 49 The close relationship between the designs in Figs. 309a and b indicates a likely connection between the geometric patterns in the northeast dome chamber in Isfahan and this anonymous treatise. Figure 309c is very similar to the pattern in Fig. 309a accept that the angular openings of the crossing pattern lines are more acute, and the six-pointed stars have been arbitrarily modified with a sixfold star rosette. This is a Zangid design from the Nur al-Din Bimaristan in Damascus (1154).

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Fig. 309

A distinct group of nonsystematic geometric designs are created from underlying tessellations that place nonagons at the vertices of a regular hexagonal grid. This variety of geometric design is most frequently found in the ornament of the Seljuk Sultanate of Rum. Figure 310 illustrates the least complex example of such a generative tessellation, with the nonagons in edge-to-edge hexagonal contact. This arrangement of underlying nonagons creates six-pointed star interstice regions at the centers of the hexagonal repeat units. This very simple underlying tessellation produces an acute pattern with six-pointed stars within the interstice regions and nine-pointed stars at the vertices of the hexagonal repetitive grid. This design was produced during the Seljuk Sultanate of Rum and is found in the triangular pendentives that support the dome at the Alaeddin mosque in Konya (1218-28). The design in Fig. 311 breaks with the repetitive convention of placing the underlying nonagons at the vertices of the hexagonal grid. Figure 311a demonstrates how the nonagons are placed at the vertices of a rhombic grid with 60° and 120° included angles. The polygonal connective matrix is comprised of triangles and shield-shaped ditrigons. Figure 311b shows how this creates an unusual median pattern comprised of nine-pointed stars that all share the same directional orientation, and it is due to this orientation that the translation symmetry is rhombic rather than hexagonal. Ignoring the arbitrary chirality of the interweaving pattern lines in Fig. 311c, this pattern adheres to the p3m1 plane symmetry group, and is one of the more interesting examples of an Islamic star pattern based upon this relatively uncommon symmetry group. Artist from the Seljuk Sultanate of Rum incorporated this design into the mosaic ornament of the Great Mosque of Malatya (1237-38). The three designs in Fig. 312 are created from an underlying tessellation that separates the nonagons at each hexagonal vertex with a ring of pentagons. This arrangement produces underlying six-pointed star interstice regions that are identical to those in Fig. 309. Each of these three examples is an acute pattern, and they only differ in the angular openings of their crossing pattern lines, and the applied pattern lines associated with the underlying six-point star interstice region. Figure 312a shows a Mamluk design from the Sultan Qaytbay complex in Cairo (1472-74) [Photograph 51]. Figure 312b is found at the Alay Han near Aksaray (1155-92) [Photograph 43], the Huand Hatun in Kayseri, Turkey (1238), as well as the Agzikara Han near Aksaray, Turkey (1242-43). The angular openings of the crossing pattern lines in this example are determined by the incorporation of regular heptagons (yellow). These heptagons are achieved in an identical fashion as the design in Fig. 309a from the northeast dome chamber in the Friday Mosque at Isfahan (1088-89). This unusual heptagonal design feature is unique to these two examples, and it would appear likely that they share a causative agent rather than having a fully independent origin. Yet their respective origins span 150 years over a distance of approximately 2000 km. The occurrence of this identical design feature over such disparate time and location may have resulted from the use of inherited pattern scrolls. The design in Fig. 312c is from one of the triangular pendentives at the Alaeddin mosque in Konya. This design eliminates the central hexagon within the pattern matrix, producing a motif that is conceptually identical to that of the comparable region in Fig. 309b. The obtuse pattern in Fig. 313a is created from the same underlying tessellation as the examples in Fig. 312. This has an arbitrary pattern line treatment within the central six-pointed star interstice region. This obtuse design was used by Shaybanid artists at the Kukeltash madrasa in Bukhara (1568-69) [Photograph 83], and the Tilla Kari madrasa in Samarkand (1646-60), and by the Janids at the Nadir Diwan Beg madrasa and khanqah in Bukhara (1622). Figure 313b shows a very successful two-point pattern also created from this underlying tessellation. This is from a Mamluk stone mosaic panel in the entry portal of the Ashrafiyya madrasa in Jerusalem (1482). Figure 313c shows an acute pattern created from a variation of this underlying tessellation that clusters six contiguous barrel hexagons around a central regular hexagon. The applied pattern lines associated with the underlying barrel hexagons and pentagons are a corollary of similar acute pattern features in the fourfold system B [Fig. 172b]. However, unlike the fourfold example, the generated octagons are not regular—although they appear to be. This example is also from the Seljuk Sultanate of Rum, and is found at the Izzeddin Kaykavus hospital and mausoleum in Sivas (1217). Figure 314 illustrates the construction sequence for the radii matrix that creates the underlying tessellation responsible for the patterns in Figs. 312 and 313a and b. Step 1 places 18 radii at each vertex of the regular hexagonal repeat unit. Step 2 establishes the edges of the nonagons, as well as the separating pentagon with the placement of a circle that is tangent to the red radii and centered on the vertex of the blue radii. Step 3 completes the nonagons and pentagon. Step 4 rotates these around the hexagon. Step 5 creates the six pentagons that surround the central six-pointed star, and Step 6 shows the complete underlying tessellation.

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Fig. 310

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Fig. 311

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Fig. 312

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Fig. 313

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Fig. 314

Figure 315 illustrates another geometric pattern with nine-pointed stars placed upon the vertices of the regular hexagonal grid. Barrel hexagons rather than the mirrored pentagons in Figs. 312 and 313 separate the nonagons within the underlying generative tessellation. This underlying tessellation has a large irregular dodecagonal interstice region at the center of each hexagonal repeat, and the applied pattern lines into this region are partially determined by the arbitrary placement of regular octagons within the pattern matrix. This pattern was produced by artists during the Seljuk Sultanate of Rum and is from the Gök madrasa and mosque in Amasya, Turkey (1266-67). Figure 316 demonstrates how the same radii matrix as that of Fig. 314 also produces the underlying tessellation for the design of Fig. 315. Step 1 draws a line that connects the red and blue radii as shown. Step 2 mirrors this line on the red radius. Step 3 draws a circle that is tangent to these reflected lines, and tangent to the blue radii. This establishes the edge for the nonagon, and the proportions for the pentagon. Step 4 completes the nonagon and pentagon. Step 5 mirrors the pentagons, and rotates these elements around the hexagon. Step 6 illustrates the completed tessellation. A notable feature of this tessellation is the ring of 12 pentagons that surround the irregular dodecagon. The irregularity of the dodecagon can be corrected through a different constructive sequence of the radii matrix [Fig. 348], thereby providing for 12-pointed stars within the completed design [Figs. 346 and 347].

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Fig. 315

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Fig. 316

As in the fivefold system, the underlying pentagons of nonsystematic patterns can be truncated into trapezoids that produce the dart motif within the acute pattern family. The design in Fig. 317 (by author) is just such a variation of the acute example in Fig. 312a. This underlying tessellation truncates the set of six pentagons that define the underlying six-pointed star so that a large underlying hexagonal interstice region is created. The lines of the dart motif inside each underlying trapezoid extend into the underlying hexagon to make the six-pointed star rosette at the center of each repeat unit. The construction of the central six-pointed star follows the standard practice for creating additive star rosettes [Fig. 222], with the point of rotation being the vertex of the underlying pentagon, two trapezoids, and large hexagon. It is somewhat surprising that this design modification does not appear to have been used historically.

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Fig. 317

Figure 318 shows a median pattern from the Great Mosque of Malatya (1237-38) that places nine-pointed stars at the vertices of the regular hexagonal grid, and six- and seven-pointed stars within the pattern matrix. Figure 318a illustrates the unusual characteristic of six edge-to-edge irregular heptagons in rotation around a central hexagon. The noticeably larger scale of the central hexagon and surrounding heptagons, relative to the length of the polygonal edges and size of the nonagons is unusual, and is responsible for the noticeable change in the design density between the peripheral and central regions of each repeat unit. The denser region of his geometric pattern shares some of the visual characteristics of median patterns from the fourfold system A [Fig. 145]. This is due to the similarity of the pentagonal and elongated hexagonal cells within the underlying tessellation and the application of approximate 90° crossing pattern lines. Interestingly, the less dense region shares the qualities of the classic median pattern with 90° crossing pattern lines created from the underlying tessellation of just regular hexagons [Fig. 95c]. Figure 319 demonstrates the construction of the underlying tessellation used to create the design in Fig. 318. This utilizes the same radii matrix as the previous examples that allows for the placement of nonagons on the vertices of the hexagonal grid. Step 1 of this more complex tessellation places intersecting perpendicular lines at the midpoint of one of the repetitive edges. In Step 2, lines are added that are perpendicular to the repetitive edge, and connect the crossing lines of Step 1 to the nearby red radii, thereby creating a hexagonal region. Step 3 mirrors one of the lines from Step 2, and uses these mirrored lines to establish the radius of a circle, which in turn, locates the edge of the nonagon. Step 4 mirrors the information from Step 3, and Step 5 rotates this information around the hexagonal repeat. Step 6 completes the tessellation by introducing irregular heptagons surrounding the central hexagon. It is important to note that while this specific construction results in a polygonal tessellation that successfully creates the median design from the Great Mosque of Malatya, as patterns and underlying tessellations become more complex, a degree of conjecture is required in establishing a construction sequence that allows for the creation of the underlying tessellations from radii matrices. The step-by-step examples provided herein are therefore not to be regarded as necessarily those used by artists of the past, but rather workable procedures that accurately replicate the underlying structures that provide for the design of the many examples of particularly complex patterns. Allowance should be made for other construction sequences that utilize radii matrices and arrive at the same polygonal result.

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Fig. 318

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Fig. 319

The designs in Fig. 300 demonstrate the wide diversity of nonsystematic patterns with 12-pointed stars as the single primary star form. A modified version of the underlying tessellation that creates these patterns produces the acute pattern in Fig. 320. This modification results from the truncation of the three edge-to-edge pentagons at the center of each triangular repeat. These truncated pentagons become three trapezoids that are contiguous with a central equilateral triangle. This acute design was used widely by many Muslim cultures. Figure 320a illustrates the truncation of the three pentagons from the underlying tessellation pictured in Fig. 300a, and Fig. 320b demonstrates the application of acute pattern lines into this modified underlying tessellation. As with other acute patterns, these trapezoids produce the distinctive dart motif. Note: the pattern lines of the obtuse angle of each dart are not located precisely on the midpoint of the truncating polygonal edge, but slightly inside the trapezoid. Different historical examples of this pattern treat this condition differently. The pattern line application of this example is determined by the decision to have parallel lines within the 12-pointed stars, and a regular hexagon within the underlying triangles. Early examples of this very popular design include the archivolt of the Zangid mihrab in the Upper Maqam Ibrahim at the citadel of Aleppo (c.1214); the vertical side panel of the Mengujekid minbar in the Great Mosque of Divrigi (1228-29); and the Seljuk Sultanate of Rum carved stone ornament at the Çifte Minare madrasa in Sivas, Turkey (1271). The seven designs in Fig. 321b–h are derived from another isometric underlying tessellation with a similar arrangement of three edge-to-edge pentagons at the center of each repetitive triangle, but with a barrel hexagon that separates each dodecagon [Fig. 299]. The three designs in Fig. 321j–l are produced from the underlying tessellation in Fig. 321i that truncates the three central pentagons in the same fashion as the example in Fig. 320. The acute pattern in Fig. 321b was used in multiple locations, including the kiosk of the Keybudadiya at Kayseri (1224-26); the late Abbasid Palace of the Qal’a in Baghdad (c. 1220) [Photograph 28]; and a Mamluk door at the Al-Azhar mosque in Cairo. A tiled variation of this acute design was used together with a square repetitive cell with matching edge conditions to create a very attractive type B dual-level design from the Topkapi ScrollFootnote 50 [Figs. 458 and 459]. The design in Fig. 321c is the standard median pattern, and that of Fig. 321d is a modification that introduces a star rosette place within the 12-pointed star in the manner that was especially popular among Mamluk artists. Surprisingly, neither of these median designs (by author) appears to have been used historically. The obtuse pattern in Fig. 321e was used as the triangular component of a Timurid type B dual-level design from the Friday Mosque at Isfahan. This dual-level design combines this triangular pattern with the square pattern in Fig. 379f. The example in Fig. 321f modifies this isometric obtuse design in Fig. 321e by adding a 6-pointed star motif into the otherwise 12-pointed stars, and subtracting the small hexagonal pattern feature at the center of each triangular repeat, thereby opening up the most congested region of the original. The overall effect of these arbitrary modifications is a field pattern with more evenly balanced pattern elements. This design was used at the Imamzada Darb-i Imam in Isfahan (1453). This same essential design, without the removed central hexagons, was used in the stone window grilles of the Sultan Qala’un funerary complex in Cairo (1284-85). Figure 321g shows the standard two-point pattern (by author) that does not appear to have been used historically; and the design in Fig. 321h employs the arbitrary modification of the 12-pointed stars through the introduction of the 12-pointed star rosette. As mentioned, this form of modification was particularly popular among Mamluk artists, and indeed, this design is found in several Mamluk locations, including: a panel from the minbar stair rail, as well as one of the interior wooden doors at the Sultan al-Mu’ayyad Shaykh complex in Cairo (1415-22); the portal of the Ribat Khawand Zaynab in Cairo (1456); and a stone lintel for a window at the Ashrafiyya madrasa in Jerusalem (1482). The modified underlying tessellation in Fig. 321i allows for the production of the well-known design in Fig. 321j, and examples include one of the Ilkhanid ceiling vaults of the mausoleum of Uljaytu in Sultaniya, Iran (1307-13); a Mamluk stone mosaic panel from the Amir Aq Sunqar funerary complex in Cairo (1346-47); and the doors of a Mamluk cupboard at the Sultan Qansuh al-Ghuri complex in Cairo (1503-05). Figure 321k shows the standard median pattern (by author) created from this modified underlying tessellation, and although this design is not known to the historical record, the variation in Fig. 321l was used in the late Abbasid tomb tower of Umar al-Suhrawardi in Baghdad (1234). The acute pattern from this modified underlying tessellation can also be created from the 3.122 underlying tessellation of triangles and dodecagons [Fig. 108d].

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Fig. 320

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Fig. 321

Figure 322 illustrates the derivation of a Timurid median pattern with 18-pointed stars placed at the vertices of the isometric grid and an arbitrary 6-pointed star at the center of each triangular repeat. Ordinarily, the outward extension of the lines from the tips of the six-pointed stars would be collinear. As a widened line or interweaving line treatment, this configuration would be less acceptable to the aesthetics of this design tradition. However, by treating this example as a tiling, the unconventional character of the noncollinearity becomes a distinctive and appealing feature within the design. This isometric pattern is from the Abdulla Ansari complex in Gazargah near Herat, Afghanistan (1425-27) [Photograph 84]. Patterns with 18-pointed stars are surprisingly uncommon, and the construction of this example is relatively simple. The underlying tessellation in Fig. 322a is made up of 18-gons placed at each vertex of the triangular repeat. These are separated by barrel hexagons along the edges of the repeat. The central region of the triangular repeat is an interstice six-pointed star that is divided into six quadrilateral kite shapes with bilateral symmetry. The angular openings of the applied median pattern lines in Fig. 322b are determined by their 18-s6 placements within the 18-gons. The underlying tessellation that creates this median pattern is well suited to the other three pattern families. Figure 323 illustrates these three additional designs (by author). Figure 323a shows an acute pattern, Fig. 323b illustrates an obtuse pattern, and Fig. 323c shows a two-point pattern. This underlying tessellation can be modified to produce yet more patterns. The three nonhistorical median patterns (by author) in Fig. 324 are created by changing the central sixfold region of the underlying tessellation in the previous examples. Figure 324a incorporates an underlying central ditrigonal element surrounded by six pentagons, Fig. 324b replaces the six pentagons with six trapezoids and a central triangle, and Fig. 324c places six pentagons around a central regular hexagon. Figures 324a and b have tight regions in the pattern matrix that are problematic, but the design in Fig. 324c is very acceptable. Figure 325a demonstrates the simple construction for the radii matrix that produces the median pattern in Fig. 322. A determination for the size of the 18-gon is shown in the detail, wherein an angle between a radius and an applied line that is perpendicular to the edge of the triangular repeat is bisected, thus establishing the placement for the edge of both the 18-gon and barrel hexagon. This construction produces a polygonal matrix between the 18-gons with edges that are congruent with the blue radii. Figure 325b introduces an alternative underlying tessellation created from this same radii matrix. This follows the previously demonstrated construction of pentagons created from circles that are tangent with the radii. The resulting ring of pentagons that surround each 18-gon are tangent with the red radii. Although this underlying tessellation does not appear to have been used within the historical record, it is fully in keeping with the expected characteristics for creating good patterns, and, as demonstrated in Fig. 326, will produce acceptable patterns in each of the four pattern families (by author). Figure 326a shows the acute pattern created from this new underlying tessellation; Fig. 326b shows the median pattern; Fig. 326c shows the obtuse pattern; and Fig. 326d shows the two-point pattern.

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Fig. 322

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Fig. 323

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Fig. 324

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Fig. 325

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Fig. 326

Figure 327 demonstrates the origins of a particularly beautiful nonsystematic two-point design with 24-pointed stars placed at the vertices of the isometric grid. Figure 327a shows how the polygonal matrix surrounding the underlying 24-gons is comprised of rings of 24 pentagons, two varieties of hexagon, and heptagons. The two-point pattern line application is testament to the ingenuity of the artist who created this design. The pattern lines within each of the clustered hexagons are particularly interesting. Each is provided with two perpendicular axis of reflected symmetry rather than the more conventional sixfold rotational symmetry commonly found within regular hexagons. This design is the product of artists working under the Seljuk Sultanate of Rum. The earliest known example is from a carved stone relief panel at the Nalinci mosque in Konya (1255-65). A later cut-tile mosaic example is from the mihrab niche at the Esrefoglu Süleyman Bey mosque in Beysehir, Turkey (1296-97) [Photograph 44]. An additional example is from a thirteenth-century stone fragment found in the Alaeddin Hill excavations in Konya.Footnote 51 Figure 328 shows three patterns (by author) created from the same underlying tessellation as shown in Fig. 327a. Although not known to the historical record, each of these is very acceptable to the aesthetics of this ornamental tradition. Figure 328a shows an acute design, Fig. 328b shows a median design, and Fig. 328c shows an obtuse design with median characteristics. The radii matrix in Fig. 329 provides the means for creating the underlying tessellation for the patterns in Figs. 237 and 238. This radii matrix places 48 radii at each vertex of the triangular repeat. Step 1 identifies an internal angle of 135°, and draws a line from the center of the triangle to the midpoint of the triangular edge. The 135° angle is close to the included angles of a heptagon. Based upon this observation, Step 2 draws 14 radii at the designated intersection of the red radii. Step 3 places a regular heptagon centered at the same intersection. The edge that is parallel to the triangular edge of the repeat rests upon the intersection of the two 14-fold radii and the two 48-fold red radii. The fact that the edges of the heptagon do not quite align with the red and blue radii is acceptable. Step 4 mirrors the lines of the heptagon as shown, and adds a line connecting the heptagon with the center of the triangular repeat. Step 5 places circles that are tangent with the red radii, and draws lines that are perpendicular to the red radii where they intersect with the circles. Step 6 uses these lines to determine the size of the 24-gon and the ring of pentagons. Step 7 rotates these elements throughout the triangular repeat, and Step 8 is the complete underlying tessellation. As mentioned previously, this specific step-by-step sequence is certainly not the only method of creating the underlying tessellation from the radii matrix, and as such is presented as a representative rather than a definitive example.

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Fig. 327

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Fig. 328

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Fig. 329

3.2.2 Orthogonal Designs with a Single Region of Local Symmetry

Nonsystematic orthogonal patterns with single regions of primary local symmetry are also well known to the historical record. Whereas the n-fold primary stars of such isometric patterns will invariably be divisible by three, those of orthogonal designs will be divisible by four. Just as with isometric examples, nonsystematic orthogonal patterns with only a single variety of local symmetry are diverse in both the types of stars and levels of complexity. Among the least complex of this variety of design are those created from simple underlying tessellations that include different combinations of octagons, irregular pentagons, squares, and triangles. Figure 330 illustrates a design that is produced from what is perhaps the least complex of such underlying tessellations. This is a Ghurid acute design from the Shah-i Mashhad madrasa at Gargistan in the remote Badghis Province of northwestern Afghanistan (1176). This very simple field pattern is characterized by octagons, four-pointed stars, and unusually elongated five-pointed stars. Other than the inclusion of the five-pointed stars, this is similar in concept to a simple design created from the 4.82 arrangement of underlying octagons and squares [Fig. 124c]. Figure 331 includes two designs where the octagonal elements in the underlying generative tessellations are separated by two irregular pentagons. These pairs of pentagons touch point to point and are separated by edge-to-edge irregular triangles. The orientation of the pentagons and triangles in these two examples is rotated 90° from one another. Figure 331a shows a Seljuk acute design from the Friday Mosque at Barsian (1105). The octagons within the pattern matrix are regular, but the heptagons are only approximate. Figure 331b shows an Ilkhanid two-point additive pattern from the tomb of Uljaytu in Sultaniya (1313-14) [Photograph 68]. The underlying generative tessellation for this design is determined by placing four equilateral triangles around each octagon, and squares with edge lengths that are equal to the octagons and equilateral triangles placed vertex-to-vertex with the triangles. The irregular triangles and pentagons are a product of this polygonal arrangement. The Ilkhanids were particularly disposed toward achieving greater geometric complexity through additive pattern elements. While almost all of their additive patterns were derived from one or another of the generative systems, this example is unusual in that the initial pattern is nonsystematic, albeit relatively simple. Figure 332 illustrates three designs created from the same underlying tessellation that places octagons at the vertices of the orthogonal grid, separated by two edge-to-edge regular hexagons along each edge of the repeat. The interior region of this polygonal arrangement is filled with irregular pentagons and heptagons, with a square at the center of the repeat. The seven-pointed stars of the median pattern in Fig. 332a have bifold symmetry and are distinctly non-regular. Several historical median designs were produced from this underlying tessellation, the earliest of which is a tiled version from the Seljuk ornament within the muqarnas of the mihrab in the Friday Mosque at Barsian in Iran (1105). The design in Fig. 332b is an interweaving version produced during the Seljuk Sultanate of Rum for the Great Mosque of Niksar, Turkey (1145). This was also used by Qara Qoyunlu artists in the portal of the Great Mosque in Van in Turkey (1389-1400). The variation in Fig. 332c is from the Ildegizid exterior façade of the Mu'min Khatun in Nakhichevan, Azerbaijan (1186). This differs from the earlier example from Naksar in the continuous pattern line treatment of the six-pointed stars located along the edges of the square repeat. The design in Fig. 332e is from the Amir Sarghitmish madrasa in Cairo (1356). Figure 332d demonstrates the unusual character of the pattern line application. With the exception of the underlying square, star rosette motifs inhabit each of the underlying polygons. As applied to the underlying heptagons and pentagons, the outer petals of the star rosettes provide a vehicle for overcoming the irregularity of these two underlying polygons. This allows for the central five-pointed stars inside the underlying pentagons, and the seven-pointed stars inside the underlying heptagons to have accurate rotation symmetry. By contrast, the multiple five- and six-pointed stars located at the vertices of the underlying tessellation in Fig. 332d are noticeably irregular. The separation of the underlying octagons with regular hexagons is similar in concept to the nonsystematic underlying tessellation in Fig. 178.

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Fig. 330

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Fig. 331

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Fig. 332

The two designs in Fig. 333 are somewhat unusual in that while their underlying tessellation has two primary polygons, the derived patterns only utilizes one of these for the primary star form. In each case, the underlying dodecagons are used to create the 12-pointed stars, while the underlying octagons contribute less overtly to the completed designs. The pattern in Fig. 333a is Ayyubid, and was used on the wooden mihrab (1245-46) of the Halawiyya mosque and madrasa in Aleppo. This makes use of 60° crossing pattern lines placed within the dodecagons in a 12-s4 arrangement that conforms to the median pattern family. The lines of the 12-pointed star are extended into the octagon and interstice regions until they meet with other extended lines. This pattern is then provided with additive six-pointed stars within the dodecagons, and additive four-pointed stars within the octagons. The design in Fig. 333b employs the same 12-pointed star within the dodecagons, but only extends the pattern lines into the interstice region. To great aesthetic effect, this design places octagons at the center of each underlying octagon. The size of these added octagons is determined by a 1/4 division of four of the underlying octagonal edges, as per two-point patterns. However, the size of these elements may also have been an arbitrary decision based upon visual effect. This design was used in two late Abbasid building in Baghdad: the Palace of the Qal’a (c. 1220) and the Mustansiriyah madrasa (1227-34).

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Fig. 333

As with nonsystematic isometric patterns, there are a relatively large number of orthogonal designs that employ 12-pointed stars as the single primary star form. The underlying tessellation in Fig. 334a is made up of dodecagons and squares in an orthogonal arrangement, with interstice regions that are divided into four pentagons. The acute design created from this tessellation has 45° crossing pattern lines placed at the midpoints of each underlying dodecagonal and square edge. The size of the regular octagon located at the center of the square repeat is determined by its extended lines being collinear with the crossing pattern lines that extend from the underlying squares into the adjacent pentagons. The sets of 45° crossing pattern lines within each dodecagon in Fig. 334a do not produce parallel lines within the 12-pointed stars. Rather, the artist responsible for this design chose to use the 30° angular openings in Fig. 334b. This subtle adjustment produces 12-pointed stars with parallel lines. The resulting Mamluk design represented in Fig. 334c was used for a pierced stone window grille at the Sultan Qala'un in Cairo (1284-85) [Photograph 55]. In addition to aesthetic preference, it is possible that the artist preferred the smaller sized 12-pointed stars as this would provide greater structural integrity within the pierced stone, as well as a more uniform light diffusion. Figure 334b shows how the crossing pattern lines of the 12-pointed stars have been moved inward from their dodecagonal edges, resulting in an elongation to the points of the five-pointed and four-pointed stars that share these edges. Moving the crossing pattern lines inward from the underlying polygonal edges is an arbitrary process aimed at achieving either an aesthetic or practical result. In this case, whereas the use 30° crossing pattern lines provides both the visual appeal of parallel lines within the 12-pointed stars and a reduced central 12-pointed star, their slight movement inward from the dodecagonal midpoints elongates the points of the 4- and 5-pointed stars in a somewhat atypical fashion, with the fourfold symmetry of the 4-pointed stars being sacrificed. Figure 335 demonstrates the derivation of four acute patterns created from an underlying tessellation comprised of dodecagons and two varieties of pentagon. The pattern in Fig. 335a is from an arched tympanum in a Seljuk gate in the Friday Mosque at Isfahan (after 1121-22). This uses 30° crossing pattern lines at each underlying polygonal edge. However, this angular opening creates a box-like octagon where the four underlying pentagons meet at the center of each repeat unit. This box-like octagon is not especially appealing and is significantly larger that the other pattern elements in this design. The artist who created this design modified the octagons with a stellated cruciform motif that is atypical to this tradition, but nicely overcomes the visual imbalance of the box-like octagon. This example of pattern adjustment is illustrative of the experimental approach to design that was taking place during the formative period when this example was produced, and it is perhaps not coincidental that these ten-sided atypical stellated motifs are very similar to the nine-sided stellated motifs used in the fivefold hybrid design in the nearby northeast dome chamber [Fig. 261]. The acute design in Fig. 335b replaces the boxlike octagon in Fig. 335a with a regular octagon with 45° crossing pattern lines. This is a very-well-known design that was used by many Muslim cultures in many locations. A particularly early example was used during the Seljuk Sultanate of Rum at the Great Mosque of Siirt (1129). The variation in Fig. 335c increases the angular opening of selected crossing pattern lines of the five-pointed stars from 30° to 60°. As with the example in Fig. 334, this example also moves the 30° angled crossing pattern lines of the 12-pointed stars inward from the midpoints of the underlying dodecagons, thereby extending the length of the points in the adjacent five-pointed stars. This example is from the Hall of the Ambassadors at the Alhambra (fourteenth century). The example in Fig. 335d uniformly employs 45° crossing pattern lines at the midpoints of each underlying polygonal edge. This design is from a frontispiece of the 30 volume Quran written and illuminated by ‘Abd Allah ibn Muhammad al-HamadaniFootnote 52 (1313) at the likely behest of the Ilkhanid Sultan Uljaytu.Footnote 53 These examples of nuanced pattern line application result in the distinct visual character of each of these four otherwise very similar designs. Figure 336 illustrates four additional designs created from the same underlying tessellation as that of the previous example. Figure 336a shows a median pattern that places central octagons with 45° crossing pattern lines into a pattern matrix that is otherwise comprised of 60° crossing pattern lines. This is an Artuqid design from the mihrab of the Great Mosque of Silvan (1152-57). Figure 336b shows a median pattern that uses 60° crossing pattern lines uniformly, but includes the customary median modification that changes the quality of the primary stars through eliminating the crossing pattern lines that are located at the midpoints of the primary polygon—in this case the dodecagons [Fig. 223]. This design can also be created from the 3.4.3.12-3.122 tessellation of decagons, squares, and triangles of the system of regular polygons [Fig. 113c]. This is a relatively common design, and a fine Muzaffarid example is from the Friday Mosque at Kerman (1349). The obtuse pattern in Fig. 336c is visually similar to a median pattern that can also be created from the 3.4.3.12-3.122 tessellation [Fig. 113a]: the difference being in the small variation between the angles of the pattern lines. While the angular variation is small, the visual results are very apparent: the design in Fig. 336c includes regular superimposed octagons, whereas the example from the system of regular polygons sacrifices these octagons in favor of the regular squares within the pattern. The superimposed octagons are an appealing feature and it is somewhat surprising that the median design in Fig. 336c (by author) does not appear to have been used historically. Figure 336d shows a two-point pattern that incorporates an octagon at the center of the repeat unit where four clustered underlying pentagons meet. This example also employs one of the more common variations to the primary stars, in this case 12-pointed [Fig. 225b]. This pattern was used in the minbar of the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81). Figure 337 illustrates the derivation of a particularly popular acute design that was used throughout Muslim cultures. This is created from a modification of the underlying tessellation in Figs. 335 and 336 that truncates the four pentagons at the center of the repeat unit, turning them into trapezoids. This produces a square at the center of the repeat unit that is contiguous with the long edge of each trapezoid. One of the earliest uses of this design in on the Ildegizid façade of the Mu’mine Khatun mausoleum in Nakhichevan, Azerbaijan (1186). Later examples include the door of the Zangid entry portal at the otherwise Mamluk Bimaristan Arghun in Aleppo; the Mamluk entry doors and incised stonework at the Zahiriyya madrasa and mausoleum of Sultan al-Zahir Baybars in Damascus (1277-81); a pair of bronze doors from the Seljuk atabeg of Cizre, Turkey (thirteenth century); the minbar at the Mamluk funerary complex of Sultan al-Zahir Barquq in Cairo (1384-86); and the minbar doors at Amir Taghribardi funerary complex in Cairo (1440). Figure 338 demonstrates a construction sequence for the underlying tessellation in Figs. 335 and 336. Step 1 places 24 radii at each vertex of the square repeat unit. Step 2 draws a circle that is tangent to the red radii. This circle is used to determine the edges of the dodecagons, as well as the separating pentagon. Step 3 completes the dodecagons and pentagon. Step 4 rotates these throughout the square repeat unit. Note: the underlying tessellation in Fig. 337 can be produced at this stage by simply connecting the four inward facing points of the four pentagons, thereby creating four trapezoids surrounding a central square. Step 5 determines the four clustered pentagons at the center of the repeat, and Step 6 shows the completed tessellation.

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Fig. 334

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Fig. 335

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Fig. 336

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Fig. 337

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Fig. 338

The underlying tessellation in Fig. 339 is unusual in that it surrounds each dodecagon with a ring of trapezoids. In this case, these trapezoids are not modifications of previously placed pentagons. Although the proportions differ, the arrangement of four trapezoids and two triangles located at the midpoints of each square repeat is conceptually identical to the 1/10 arrangement of similar modules from the fivefold system [Fig. 196]. In both cases, this arrangement lends itself to the acute pattern family, and the similarity in the resulting concave octagonal pattern motif centered in this arrangement is readily apparent when compared to certain historical fivefold designs, for example, a Timurid design from the Shah-i Zinda in Samarkand [Fig. 254b]. The underlying tessellation for the design in Fig. 339 is essentially corner-to-corner dodecagons placed upon the vertices of the orthogonal grid with four equilateral triangles placed at the center of the square repeat unit. This creates the elongated rhombic interstice regions shown as two mirrored triangles. While the acute pattern that this tessellation creates is attractive, the irregular eight-pointed star at the center of the repeat unit is improved by adding four more points, thereby making this into a regular 12-pointed star as per Fig. 340a. As mentioned previously, the geometric design tradition in the Maghreb placed particular emphasis on the arbitrary modification of the primary star forms.Footnote 54 In addition to the arbitrarily modified non-regular 8-pointed stars that have been given four more points so that they become 12-pointed stars, the examples in Fig. 340b–d illustrate three distinctive variations to the primary 12-pointed stars. Each of these four variations has its own distinct aesthetic merit, and each of these four examples was used in the zillij cut-tile mosaics of the Alhambra (fourteenth century). In fact, each was used within the single zillij panel represented in Fig. 341. The incorporation of multiple varieties of modified 12-pointed stars into a single panel is an effective means of bringing broad-ranging design variation into what would otherwise be a more repetitive aesthetic.

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Fig. 339

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Fig. 340

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Fig. 341

The underlying tessellation in Fig. 342 is unusual in that it is comprised principally of rings with 12 irregular heptagons that surround a 12-pointed star interstice region at each corner of the orthogonal grid. This is an Ilkhanid median design from the mausoleum of Uljaytu in Sultaniya, Iran (1307-13) [Photograph 87]. The heptagons in the underlying tessellation are of two varieties, and they produce a very distinctive design characterized by twelve 7-pointed stars surrounding 12-pointed stars. The four underlying heptagons at the center of each repeat unit create a motif of four point-to-point seven-pointed stars. Figure 343 demonstrates a construction of the underlying tessellation for this design. This involves the placement of 12 regular heptagons in 12-fold rotation upon the vertices of a dodecagon as shown, as per Step 1. Detail 1 shows how the midpoints of the heptagonal edges intersect with the midpoints of the dodecagonal edges, but the edges of adjacent heptagons overlap rather than being contiguous. Step 2 places this ring of heptagons at each corner of the square repeat unit. This conceptually completes the tessellation. However, the nonaligned edges of the adjacent heptagons need to be corrected to make the pattern line application easier. Detail 2 shows how the edges of the heptagons are not aligned with one another. This is rectified by simply making all the irregular heptagonal edges that intersect with the formative dodecagons, triangles, and squares perpendicular with these formative edges. Step 3 shows the new set of heptagons that are slightly irregular, and have edges that are aligned with the red radii of the radii matrix. Step 4 shows the completed tessellation. This interesting tessellation will create very acceptable designs in each of the four pattern families, although no other examples are known to the historical record.

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Fig. 342

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Fig. 343

The acute design in Fig. 344d is from a Mamluk frontispiece of an illuminated Quran (1369) commissioned by Sultan Sha’ban.Footnote 55 This places 16-pointed stars at the vertices of the orthogonal grid and octagons at the center of each square repeat unit. This underlying tessellation includes an unusual feature of alternating pentagons and barrel hexagons that surround the 16-gons. Figure 344a illustrates a standard acute pattern line application to the underlying tessellation. This places a four-pointed star within the underlying square at the center of the repeat unit. The Mamluk artist who created this design chose to alter this central region. Figure 344b arbitrarily introduces an octagon surrounded by 4 eight-sided mushroom-shaped elements at the center of each repeat unit. This Quranic illumination places the central repetitive regions at the corners of the panel, allowing only 1/4 of each central region to be represented. For reasons that remain unclear, the artist chose to mirror the 1/4 segment of the octagon at each corner of the panel, thereby reversing the direction of the foot of each mushroom shape. However, this does not work well when repeating the design with translation symmetry, and has been ignored for the purposes of this study. This artist introduced the further modification of the primary 12-pointed stars shown in Fig. 344c. This conforms to the common practice for modifying the primary stars of median designs [Fig. 223], although this is applied to an acute rather than a median pattern. The design in Fig. 345 is from the same Mamluk illuminated Quran commissioned by Sultan Sha’ban,Footnote 56 and this example also places 16-pointed stars onto the vertices of the orthogonal grid. Like the underlying tessellation of Fig. 344, there are alternating barrel hexagons arranged around the 16-gon, but in this case, the barrel hexagons are rotated 90°, and separated by an unusual octagonal interstice region. Such oddly proportioned interstice regions are often problematic when applying the pattern lines. However, in this case the bilateral pattern lines work very nicely within this atypical feature. The similarity between this example and the previous pattern is not surprising in that both of these designs with 16-pointed stars were presumably the work of the same artist. Figure 345a illustrates the standard acute pattern with 45° crossing pattern lines. As with the previous example from the same Quran, Fig. 345b modifies the primary 12-pointed stars in the manner that was particularly popular among Mamluk artists [Fig. 223]. Again, this modification is typically applied to median patterns, but in this case has been applied to the acute pattern. The angles of the resulting dart motifs within this modification are consequently more acute than normally found with median patterns. Figure 345c shows four repeat units of this design in a two-color tiling expression. Figure 345d is a representation of the design as it was used in the Quranic illumination, albeit with different colorization and no floral infill.

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Fig. 344

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Fig. 345

3.2.3 Isometric Designs with Multiple Regions of Local Symmetry

There is a wide diversity of nonsystematic patterns that repeat upon an isometric grid and have more than one region of primary local symmetry. These invariably place objects with sixfold symmetry at the vertices of the triangular grid, and objects with threefold symmetry at the vertices of the dual-hexagonal grid. Additional introduced regions of local symmetry may include the midpoints along the repetitive edges, which will always have twofold point symmetry (i.e., n-pointed stars with even numbers), and occasionally locations within the field of the design. The n-fold symmetry of local regions within the field is only limited by the inherent geometry of a given construction.

There are only a few nonsystematic underlying tessellations that were used historically to create designs in each of the four pattern families. The underlying tessellation that produces the patterns in Figs. 346 and 347 is perhaps the most prolifically used, and the designs created from each of the pattern families are exceptionally beautiful. This underlying tessellation places dodecagons at the vertices of the triangular grid and nonagons at the vertices of the dual-hexagonal grid. A ring of pentagons surrounds each dodecagon, and the nonagons are separated by barrel hexagons. This configuration of polygons is the least complex of several historical underlying tessellations that were used to create patterns with 9- and 12-pointed stars. Figure 346a demonstrates the derivation of the acute pattern created from this underlying tessellation. This pattern was used widely throughout Muslim cultures, and the interweaving example in this figure represents the interpretation used on two illuminated facing pages of a Moroccan Quran written in 1568Footnote 57 [Photograph 65]. Early examples of this nonsystematic acute design include a Mengujekid carved stone border from the east portal of the Great Mosque and hospital of Divrigi (1228-29); a Mamluk bronze door from the Sultan al-Zahir Baybars madrasa in CairoFootnote 58 (1262-63); and the Mamluk ornament of the mausoleum of Fatima Khatun in Cairo (1283-84). Perhaps the most unexpected location for the use of this acute pattern is from a Mudéjar door in the Cathedral of Santo Domingo in Cusco, Peru (1559-1654). Figure 346b illustrates the median design created from the same underlying tessellation. Unlike the acute pattern, this design is uncommon, and a fine historical example is found at the mausoleum of Uljaytu in Sultaniya, Iran (1307-13). This is depicted at a much later date within one of the Tashkent scroll fragments (sixteenth or seventeenth century) at the Institute of Oriental Studies in Tashkent, Uzbekistan. Figure 347a shows an obtuse pattern created from this same underlying tessellation, and the earliest known use of this design is in the carved stucco ornament that frames the arched Fatimid mihrab at the al-Azhar mosque in Cairo. This was likely produced during the first half of the twelfth century as part of the extensive renovations of the Fatimid caliph al-Hafiz li-Din AllahFootnote 59 (r. 1131-49). As such, this example is significant in that it appears to be the earliest extant nonsystematic pattern with more than one variety of primary star to originate from outside those regions under direct Seljuk artistic influence. Another early example of this obtuse pattern is found in the mihrab of the Great Mosque at Aksehir in Turkey (1213). Locations of later examples include the Qara Qoyunlu ornament of the Great Mosque of Van (1389-1400); a Timurid cut-tile mosaic border from the Abdulla Ansari complex in Gazargah, Afghanistan (1425-27) [Photograph 85]; a pierced stone screen from the minaret balcony of the Amir Ghanim al-Bahlawan funerary complex in Cairo (1478); and the Shaybanid cut-tile mosaic arch spandrel in the entry portal of the Kalyan mosque in Bukhara (1514). The two-point pattern in Fig. 347b is also created from this same tessellation, and employs the most frequently used modification to the 12-pointed stars [Fig. 223], and the distinctive variation to the pattern lines of the 9-pointed stars [Fig. 225d]. The angles of the pattern lines within the nine-pointed stars are derived from the two-point 9-s3 pattern line application. The earliest known example of this design is Ilkhanid: from the mausoleum of Uljaytu in Sultaniya, Iran (1307-13). Several Mamluk examples include a panel from the Mamluk minbar at the Qadi Abu Bakr Muzhir complex in Cairo (1479-80); an incised stone relief panel in the entry portal of the Qanibay Amir Akhur funerary complex in Cairo (1503-04); and a contemporaneous rectangular side panel from the minbar of the Sultan Qansuh al-Ghuri complex (1503-05) [Photograph 53]. Figure 348 demonstrates a construction of the underlying tessellation that produces the patterns in Figs. 346 and 347. This begins with the placement of 24 radii at the vertices of an equilateral triangle as shown in Step 1. Step 2 introduces 18 radii at the center of the triangle, and completes the radii matrix that produces this underlying tessellation. Step 3 determines the sides of the nonagon and dodecagon, as well as the proportions of the irregular pentagons. This follows the standard procedure described earlier. These three polygonal elements are completed in Step 4. Step 5 mirrors and rotates these elements throughout the triangle, and Step 6 illustrates the completed underlying tessellation.

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Fig. 346

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Fig. 347

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Fig. 348

Figure 349 shows how the same radii matrix with 24-fold and 18-fold centers of local symmetry can be used to create another underlying tessellation that also produces fine patterns with 12- and 9-pointed stars. As in the fivefold system, the arrangement of six pentagons surrounding the thin rhombus makes this alternative underlying tessellation especially appropriate for obtuse and two-point patterns [Fig. 197]. Step 1 draws a line that connects the vertices of the red and blue radii as shown. Step 2 reflects this line upon the edge of the triangle, and upon one of the red central radii. Step 3 uses these lines to determine the edges of the nonagons and dodecagons, as well as two varieties of pentagon. Step 4 completes the nonagon and dodecagon, as well as the pentagons. Step 5 mirrors and rotates these elements throughout the triangle, thereby determining the barrel hexagons and small rhombi along each triangular edge. Step 6 illustrates the completed underlying tessellation. Figure 350a illustrates the obtuse pattern (by author) created from this underlying tessellation, and Fig. 350b illustrates the two-point pattern (by author) produced from this same underlying tessellation, and the 12-pointed stars have been modified in the standard fashion [Fig. 223]. While both of these patterns are aesthetically acceptable to this tradition, neither appears to have been used historically. As with fivefold examples with the cluster of six pentagons surrounding a thin rhombus, the underlying tessellation in Fig. 350 requires modification for the production of successful acute patterns. As explained previously [Fig. 198], there are two varieties of such modification. The acute design (by author) in Fig. 351a is created from the truncation of just four of the six clustered pentagons. This transforms the four pentagons into trapezoids that then define a wide rhombic interstice region. The acute design (by author) in Fig. 351b modifies the cluster of six pentagons by truncating all six pentagons such that they become six trapezoids surrounding a concave hexagonal interstice region. The pattern line application within the trapezoids of both these designs maintains the distinctive dart motif common to acute patterns. Although aesthetically acceptable, neither of these modified patterns is known to the historical record.

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Fig. 349

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Fig. 350

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Fig. 351

Figure 352 illustrates the derivation of another obtuse pattern with 12-pointed stars placed at the vertices of the isometric grid and 9-pointed stars at the vertices of the dual-hexagonal grid. The underlying polygonal elements and applied pattern lines have shared characteristics with many designs created from the fivefold system [Fig. 233a]. What would otherwise be 12-pointed stars within this design have been given a sixfold additive modification that is conceptually the same as the far more common fivefold modification found in obtuse patterns from the fivefold system [Fig. 224a]. As with analogous patterns created from similar underlying polygonal modules from the fivefold system, this pattern can also be created from an alternative underlying tessellation of dodecagons, pentagons, barrel hexagons, and thin rhombi (not shown). This obtuse design was produced during the Seljuk Sultanate of Rum, and is from the Sultan Han in Aksaray, Turkey (1229). Figure 353 demonstrates how the construction of the underlying tessellation for this pattern can be produced from a radii matrix that begins with Step 1 wherein a line that intersects the third vertices of the nonagon is extended outside the nonagon to a length that is equal to the sides of the nonagon. Step 2 mirrors this process as shown and places a line (red) at the ends of these extended lines that will become an edge of the triangular repeat. Step 3 mirrors the two extended lines to make the hexagonal element, and rotates the edge three times around the nonagon to make an equilateral triangle that is centered on the nonagon. Step 4 rotates the elongated hexagons to each side of the triangle. Step 5 uses the acute points of the elongated hexagons to determine the size and position of the dodecagons. And Step 6 shows the completed triangular repeat for the underlying tessellation.

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Fig. 352

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Fig. 353

Figure 354 details the construction of yet another obtuse pattern with 12-pointed stars on the vertices of the isometric grid and 9-pointed stars at the vertices of the dual hexagonal grid. Like the pattern in Fig. 352, this pattern can be made from either of two underlying tessellations. Figure 354a illustrates the pattern derivation from an underlying tessellation of dodecagons, nonagons, pentagons, barrel hexagons, and thin rhombi. Figure 354b uses a dual underlying tessellation of dodecagons, nonagons, elongated hexagons, and concave hexagons to create the same design. Figure 354c demonstrates the dual relationship between these alternative tessellations. This is analogous to the dual relationship in many examples from the fivefold system [Fig. 200]. This design is also from the Seljuk Sultanate of Rum, and comes from the Susuz Han in the village of Susuzköy, Turkey (1246).

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Fig. 354

Figure 355 shows the construction of a Mamluk median design from a window lintel in the Qartawiyya madrasa in Tripoli, Lebanon (1392). Figure 355b shows the standard median derivation for this pattern, and Fig. 355a demonstrates how this design is created from an underlying tessellation comprised of 15-gons, dodecagons, pentagons and barrel hexagons. The design in Fig. 355d is the historical example that modifies the 15-pointed stars as per the common fivefold convention [Fig. 223]. These two patterns place 12-pointed stars on the vertices of the isometric grid, and 15-pointed stars at the vertices of the dual hexagonal grid. Figure 355a shows how the pattern lines within the pentagons and barrel hexagons that surround the 15-gon are alternating five-pointed stars and darts. In Fig. 355c the pattern lines within these polygonal elements eliminate the points that extend into the 15-gons, and extend the other lines to create the ring of 15 darts that characterize this modification. Figure 356 demonstrates a construction of the underlying tessellation that produces this pattern. This process begins with the placement of 24 radii at the vertices of an equilateral triangle and 30 radii at the center of the triangular repeat as shown in Step 1. Step 2 places a line that connects the red and blue vertices as shown. Step 3 mirrors this line along the edge of the repeat. Step 4 determines the sides of the dodecagon and 15-gon as well as the proportions of two irregular pentagons. Step 5 completes these elements. Step 6 mirrors one of the pentagons, determines a third pentagon by mirroring one of the sides of the 15-gon and mirroring the adjacent radii, and establishes the barrel hexagon from the blue radii. Step 7 mirrors these elements as shown. Step 8 rotates these elements three times around the center of the triangular repeat. And Step 9 shows the completed underlying tessellation.

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Fig. 355

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Fig. 356

Figure 357 demonstrates the derivation of an acute pattern with 18-pointed stars placed upon the vertices of the isometric grid and 9-pointed stars upon the vertices of the dual-hexagonal grid. This design also has regular octagons at the midpoints of the edges of both the triangular and hexagonal repetitive cells, where these two grids intersect. This is a Mamluk design from the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81). Figure 358 demonstrates a construction of the underlying tessellation that produces this pattern. Step 1 locates 36 radii at the vertices of an equilateral triangle and 18 radii at the center of the triangular repeat. Step 2 establishes the size of the 18-gon and nonagon, as well as the pentagon that separates the 18-gon and nonagon. Step 3 completes the 18-gon and nonagon, as well as the separating pentagon. Step 4 establishes two more pentagons placed at two midpoints of the triangular repetitive edges. Step 5 mirrors these new pentagons along the sides of the triangular repeat, and rotates the polygonal elements throughout the triangular repeat. Step 6 illustrates the completed underlying tessellation.

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Fig. 357

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Fig. 358

Figure 359 shows an acute design with 18-pointed stars at the vertices of the isometric grid and 12-pointed stars at the vertices of the dual hexagonal grid. This pattern can be created from either of two underlying tessellations. Figure 359a is a simple edge-to-edge configuration of 18-gons and dodecagons, while the underlying tessellation of Fig. 359d is comprised of the same primary polygons, but with a connective polygonal matrix that includes pentagons, trapezoids, and interstice concave hexagons. Figures 359c and f are identical except for scale. This Mamluk design is from the Sultan al-Zahir Barquq madrasa and khanqah in Cairo (1384-1386) [Photograph 54]. Figure 360 shows a construction of the underlying tessellation that produces the pattern from Fig. 359. This begins with the placement of 36 radii at the vertices of an equilateral triangle and 24 radii at the center of the triangular repeat. Step 1 illustrates this radii matrix. Step 2 determines the sides of the dodecagon and 18-gon, as well as the proportions of the irregular pentagon that separates these primary polygons. This also places a line that connects the end points of blue and red radii that is a precursor for another pentagon. Step 3 completes these polygonal elements, as well as establishes the proportions for the additional pentagons and thin rhombus by mirroring the precursor line from the previous step. Step 4 mirrors these additional pentagons. Step 5 rotates these polygonal elements throughout the triangular repeat; and Step 6 completes the underlying tessellation. As with previous examples, the arrangement of six clustered pentagons surrounding a thin rhombus that occurs along each edge of the triangle is not suitable for the production of acute patterns. The acute design in Fig. 359 makes use of the modification to these clustered pentagons as per established convention [Fig. 198] wherein all six pentagons are truncated to form six trapezoids with a concave hexagonal interstice region.

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Fig. 359

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Fig. 360

As mentioned, some isometric designs with more than one variety of primary star place one of these stars at the vertices of the triangular grid and the other at the midpoints of the edges of the triangular cell. The acute design in Fig. 361 places 12-pointed stars at the vertices of the isometric grid, and 8-pointed stars at these midpoints. The application of the pattern lines allows for the introduction of nonagons at the centers of the triangular repetitive cells, which are also the vertices of the dual-hexagonal grid. This design is from the Seljuk Sultanate of Rum, and was used on one of the many gravestones at Ahlat near the shores of Lake Van in eastern Turkey (thirteenth to fifteenth centuries). The construction of the underlying tessellation that creates this design is shown in Fig. 362. This begins with setting up the radii matrix. Step 1 places 24 radii at the vertices of the triangular repeat. Step 2 introduces 16 radii at the midpoints of the edges of the triangular repeat. This completes the radii matrix. Step 3 determines the size and placement of the dodecagons and octagons within the underlying tessellation. This also identifies the proportions of the pentagons that surround the dodecagons. Step 4 completes the dodecagon, octagon, and pentagon. Step 5 mirrors the pentagon. Step 6 rotates these polygonal elements throughout the triangular repeat. Step 7 fills the interstice regions at the center of the triangle from Step 6 with three irregular hexagons with edges that are congruent with the red radii. And Step 8 illustrates the completed underlying tessellation. Figure 363 shows the construction of an obtuse pattern that places 12-pointed stars at the vertices of the isometric grid and 10-pointed stars at the midpoints of each edge of the triangular repeat units. A measure of the success of this fine Qara Qoyunlu design is that it appears less complex than it actually is. This was used in the raised brick ornament of the Great Mosque at Van in eastern Turkey (1389-1400) [Photograph 86]. The similarity and close proximity of this example to the previous design in Fig. 361, albeit with ten- rather than eight-pointed stars, suggest the possibility that they may have been produced by the same individual or atelier, or at the least, that one may have inspired the origin of the other. The construction of the underlying tessellation that produces this design is demonstrated in Fig. 364. This begins with the construction of the radii matrix. Step 1 places 24 radii at the vertices of the triangular repeat. Step 2 introduces 20 radii at the midpoints of the edges of the triangular repeat, thus completing the radii matrix. Step 3 determines the size and placement of the dodecagons and decagons within the underlying tessellation. This also identifies the proportions of the pentagons that surround the dodecagons. Step 4 completes the dodecagon, decagon, and pentagons. Step 5 mirrors the pentagon. Step 6 rotates these polygonal elements throughout the triangular repeat. Step 7 fills the interstice regions at the center of the triangle from Step 6 with three clustered pentagons at the center of the triangle surrounded by three irregular hexagons. And Step 8 illustrates the completed underlying tessellation.

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Fig. 361

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Fig. 362

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Fig. 363

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Fig. 364

The acute pattern in Fig. 365 has three regions of local symmetry, with 24-pointed stars at the vertices of the isometric grid, 12-pointed stars at the vertices of the dual hexagons grid, and 8-pointed stars at the midpoints of the edges of the triangular repetitive cells. The crossing pattern lines of the 24-pointed stars and 12-pointed stars have been arbitrarily moved slightly inward from the midpoints of their associated underlying polygonal edges toward the centers of their respective primary polygons. This allows the parallel pattern lines within the underlying octagons to be replicated within the underlying dodecagons, and closely simulated within the 24-gons. Were the crossing pattern line associated with the 24- and 12-pointed stars placed on the midpoints of their respective primary polygons, the sizes of the central 24- and 12-pointed stars would be substantially increased, thereby changing the aesthetic character of the finished design. This is a Seljuk Sultanate of Rum pattern from the mihrab niche at the Great Mosque of Ermenek in Turkey (1303). Figure 366 demonstrates a construction of the underlying tessellation for this design, beginning with the construction of the radii matrix. Step 1 places 48 radii at the vertices of the triangular repeat. Step 2 introduces 24 at the center of the repeat. Step 3 places 16 radii at the midpoints of the edges of the repeat. This completes the radii matrix. Step 4 determines the size and placement of the dodecagons and octagons within the underlying tessellation, as well as the pentagon that separates these two polygons. This also places a 24-gon of a suitable size to allow for a well-proportioned pentagon between the 24-gon and the octagon. Step 5 emphasizes the pentagons the separate the 24-gon, dodecagon, and octagon. Step 6 mirrors these polygons. Step 7 rotates these throughout the triangular repeat; and Step 8 illustrates the completed underlying tessellation.

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Fig. 365

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Fig. 366

The design in Fig. 367 incorporates a fourth region of local symmetry located within the field of the pattern matrix. This remarkable acute design has 9-, 10-, 11-, and 12-pointed stars, and is arguably the most complex nonsystematic isometric design from this ornamental tradition. Figure 367a shows how the underlying tessellation for this design places dodecagons at the vertices of the isometric grid (blue), nonagons at the vertices of the dual hexagonal grid (green), decagons at the midpoints of the edges of the triangular cells (purple), and hendecagons within the field of the polygonal matrix (grey). These primary polygons allow for the introduction of the four varieties of local symmetry to the completed design. The four primary polygons are surrounded by a connective matrix of pentagons and barrel hexagons. This masterpiece of geometric art was produced during the Seljuk Sultanate of Rum, and two examples are found within the architectural record: the carved stone ornament in the courtyard portal of the Seri Han near Avanos (1230-35), and the carved stone ornament of the entry to the mosque at the Karatay Han near Kayseri (1235-41). These contemporaneous examples are only 65 km apart, and are almost certainly the work of the same artist or atelier. The three designs with 9-, 10-, 11-, and 12-pointed stars in Fig. 368 (by author) are created from the same underlying tessellation as the previous acute pattern. Figure 368a shows a median pattern, Fig. 368b shows an obtuse pattern, and Fig. 368c shows a two-point pattern. Each of these is acceptable to the aesthetics of this tradition, but are unknown to the historical record. Collectively, these demonstrate the efficacy of the polygonal technique in creating highly complex designs in each of the four pattern families from a single underlying tessellation. The construction of the underlying tessellation for these designs is significantly more complex than previous examples. Figure 369 details the creation of just the radii matrix. This requires subtle geometric adjustments during the construction to allow for the incorporation of the regions with 11-fold radial symmetry. Step 1 places 24 radii at the vertices of the triangular repeat. Step 2 adds 18 radii to the center of the triangle. Step 3 incorporates 20 radii at the midpoint of each edge of the triangle. The maker of this pattern could have finished the radii matrix at this point—moving directly to the establishment of an underlying tessellation that will create patterns with 9-, 10-, and 12-pointed stars. However, a close examination of the radii matrix in Step 3 reveals the potentiality of the further incorporation of local 11-fold symmetry. The introduction of the 11-fold region begins with Step 4, which highlights the 114° and 66° angles of the radii from Step 3. As further shown, these are very close to the 114.5454…° and 65.4545…° angles found in 22-fold rotational symmetry, and indicate that regions with 11-fold symmetry might successfully be incorporated into this radii matrix. Step 5 places 22 radii at the intersection of the radii from the 24-, 20-, and 18-fold centers. Detail 1 shows how the line of radius from the 20-fold center does not quite intersect the 22-fold center; and conversely, the line of radius from the 22-fold center does not quite intersect with the 20-fold center. Detail 2 introduces a vertical line half way between these two centers. Detail 3 moves the 22 radii vertically by the small amount so that the two lines of radius from the 20- and 22-fold centers intersect. This detail also trims these lines. These two radii now appear collinear, but actually have a slight angle off 180°. Step 6 rotates the region with 22-fold symmetry into the other two positions within the triangle. The small dots indicate the radii intersections that appear collinear. Figure 370 demonstrates the construction of the underlying tessellation from the radii matrix in Fig. 369. This follows established procedure of using tangent circles to determine the pentagons and primary higher order polygons. Step 1 determines the edge size and location for the nonagon, decagon, and hendecagon through placing circles at the vertices of the red radii that are tangent to the blue radii. Step 2 completes these three primary polygons. Step 3 emphasizes the various pentagons and barrel hexagons that were also created from the tangent circles that separate the primary polygons of Step 2. Step 3 also determines the dodecagon by drawing a line that connects red and blue radii, thereby creating a pentagon, and drawing a circle that is tangent with this pentagon and the blue radii. This circle is centered on the edge of the triangular repeat unit. Step 4 completes the dodecagon, and emphasizes the additional pentagons and barrel hexagons implicit within the radii matrix. This step also mirrors the polygonal elements. Step 5 rotates the polygonal elements from Step 5 throughout the triangular repeat; and Step 6 shows the completed underlying tessellation.

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Fig. 367

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Fig. 368

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Fig. 369

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Fig. 370

Clearly, Muslim artists were immensely innovative in developing repetitive strategies that allowed for the use of multiple centers of local symmetry within a single pattern. However, they by no means exhausted the pool of repetitive polygonal stratagems that provide for the creation of visually compelling geometric designs. An exploring mind can discover new forms of underlying polygonal repetitive stratagems that are effective, but do not appear within the historical record. These can provide contemporary artists with tremendous potentiality for the creation of original nonsystematic geometric designs, often imbued with considerable complexity. Within the isometric family, there are at least two contemporary repetitive stratagems that are particularly interesting, and worthy of detail. Both of these will produce a very large number of very successful patterns, and can be regarded as doorways into uncharted creative territory.

Figure 371 illustrates the principle behind the first of these contemporary methods for producing underlying generative tessellations. This type of polygonal construction is governed by the use of a central polygon with sides that are multiples of three (e.g., triangle, hexagon, nonagon, dodecagon). Onto this central polygon are placed three edge-to-edge polygons in threefold rotation around the central polygon. Lines are drawn that bisect these three new polygons. These three lines are extended until they meet, thus creating an equilateral triangle. The triangle and polygonal elements are mirrored to create a rhombus that is then rotated three times to create a regular hexagonal repeat unit with translation symmetry. Patterns based upon this repetitive schema are of the p31m plane symmetry group. In addition to the author, both Peter Cromwell in the United Kingdom, and Goossen Karssenberg of the Netherlands independently discovered this method of constructing polygonal matrices.Footnote 60 The examples in Fig. 371 are just eight such tessellations, but very many more are possible, and each of these has the potential for creating very successful patterns in each of the four pattern families. Examples A, B, C, and D all use the hexagon as the central polygon, while examples E, F, G, and H use the nonagon at the center. Example A adds three pentagons to the hexagon, creating an interstice region of three clustered irregular heptagons. Example B adds heptagons to the central hexagons. This serendipitously provides for the incorporation of dodecagons into the generative tessellation. Example C adds octagons to the central hexagons, and example D adds nonagons to the hexagons. The inclusion of the nonagons allows for the secondary placement of additional hexagons at the vertices of the repetitive grid. This tessellation makes particularly nice geometric patterns. Example E adds heptagons to the central nonagons. This tessellation also produces very nice patterns. Example F adds octagons to the central nonagons, and examples G and H both add decagons. The arrangement of nonagons and decagons in Example H allows for the serendipitous addition of 15-gons at the vertices of the rhombic repeat. Figure 372 shows the construction of a nonhistorical median pattern (by author) comprised of seven- and nine-pointed stars created from the tessellation in Fig. 371e.Footnote 61 This has shared visual characteristics to the median patterns created from the fourfold system A [for examples, Figs. 145 and 154]. Indeed, the success of this design is predictable through the principle of adjacent numbers wherein a star form that works well on its own (in this case the eight-pointed star) indicates the likely success of patterns that use stars that are one numeric step above and below this star form. This is a useful principle for predicting patterns with unexpected combinations of local symmetry—in this case seven- and nine-pointed stars. Figure 373a illustrates a more complex form of this repetitive strategy that combines central dodecagons with three edge-to-edge hendecagons (11-gons) in threefold rotation around it. As with the examples from Fig. 371, the elements from the initial triangle have been mirrored to form a rhombic cell that is then rotated three times to produce the hexagonal repeat unit with translation symmetry. Figure 373b illustrates the serendipitous addition of tridecagons (13-gons) into this configuration. However, the edges of the tridecagons are not quite congruent with their neighbors. This configuration of hendecagons, dodecagons, and tridecagons (11-, 12-, and 13-gons) can be used on its own to create patterns, but has limitations. However, as demonstrated previously, greater design potential is derived from underlying generative tessellations that separate the primary polygons with a connective matrix of secondary polygons. Figures 373c and d demonstrate how the polygonal tessellation in Fig. 373b can be used to produce a very satisfactory radii matrix from which new underlying tessellations can be created. Lines are simply introduced that connect the centers to the vertices and midpoints of each primary polygon, and extend these radii outward until they meet with other extended radii. Figure 373d shows the completed radii matrix. Figure 373e shows an underlying tessellation that can be created from this radii matrix, and Fig. 373f shows just the underlying tessellation within its hexagonal repeat unit. The initial non-congruence between the tridecagons and their adjacent hendecagons and dodecagons is eliminated through the matrix of secondary connective polygons, allowing each of the three primary polygons within this underlying tessellation to have regular n-fold symmetry. Figure 374 is an acute pattern (by author) created from the underlying tessellation constructed in Fig. 373. This design is comprised of 11-, 12-, and 13-pointed stars, with octagons also incorporated into the pattern matrix. While not historical, this acute pattern comports with the aesthetics of this tradition, especially as practiced in Anatolia during the Seljuk Sultanate of Rum. This seemingly eccentric combination of star forms is less mysterious when compared with the 3.122 semi-regular isometric tessellation of dodecagons and triangles [Fig. 89]. This places six edge-to-edge dodecagons around each individual dodecagon. The underlying tessellation for the pattern in Fig. 374 can be interpreted as having replaced these six surrounding dodecagons with alternating hendecagons (green) and tridecagons (yellow). This is another form of the principle of adjacent numbers where the original polygonal figure, in this case the dodecagon, is maintained, and the surrounding dodecagons are replaced with polygons that have alternating plus-one and minus-one number of sides. The underlying tessellation used to produce this pattern will also make very successful median, obtuse, and two-point patterns.

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Fig. 371

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Fig. 372

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Fig. 373

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Fig. 374

Figure 375 demonstrates the principle behind the second hexagonally based repetitive stratagem that, although ahistorical, nonetheless produces a very wide variety of underlying generative tessellations with tremendous potential for contemporary geometric artists. This method of creating nonsystematic patterns with unusual combinations of local symmetry makes use of two varieties of irregular hexagonal cell (red) within a single repetitive construction. The blue hexagons in Fig. 375 are centered on the isometric grid, while the beige hexagons are centered upon the dual hexagonal grid. The distribution of these two types of hexagon is similar to the active and passive underlying hexagonal grid in the patterns of Fig. 98. However, the included angles of both varieties of hexagon in Fig. 375 have been adjusted to conform to the angles associated with specific polygons: thus allowing for the incorporation of unusual combinations of primary polygons, with their associated local symmetries, into an underlying generative tessellation. The angular proportions of Fig. 375a are derived from a combination of regular pentagons and heptagons at the vertices of the irregular hexagonal grid, and regular hexagons at the centers of the beige hexagons. This will produce patterns with five-, six-, and seven-pointed stars. The alternating pentagons and heptagons around each hexagon are another example of the principle of adjacent numbers. The lack of precise edge-to-edge alignment in this configuration of polygons will cause problems when extracting patterns. As in Fig. 373, this can be overcome by using the regular polygons as a layout for a radii matrix that can then be used to create a new underlying tessellation with connecting irregular polygons separating the regular primary polygons. The proportions of the hexagonal grid in Fig. 375b are based upon an arrangement of tetradecagons (14-gons) and decagons at the vertices of the hexagonal grid, with dodecagons at the centers of the blue hexagons. This will make patterns with 10-, 12-, and 14-pointed stars. The included angles of what would otherwise be the blue hexagons in Fig. 375c are 60° and 180°: affectively turning the hexagon into an equilateral triangle. This places octagons and dodecagons into an edge-to-edge configuration that allows for patterns with 8- and 12-pointed stars. Figure 375d places octagons and decagons at the vertices of the hexagonal grid, and hexagons at the centers of the blue hexagons. This will produce patterns with six-, eight-, and ten-pointed stars. Figure 375e places hendecagons (11-gons) and tridecagons (13-gons) at the vertices of the hexagonal grid, with dodecagons at the centers of the brown hexagons. As with the example in Fig. 374, this will produce patterns with 11-, 12-, and 13-pointed stars, although with a completely different repetitive schema. Figure 375f places heptagons and octagons onto the vertices of the hexagonal grid, allowing for patterns with seven- and eight-pointed stars. Figure 375g places nonagons and decagons at the vertices of the hexagonal grid, allowing for patterns with nine- and ten-pointed stars. And Fig. 375h places decagons and hendecagons at the vertices of the hexagonal grid, with nonagons at the centers of the brown hexagons. This will produce patterns with 9-, 10-, and 11-pointed stars, again with a demonstration of the principle of adjacent numbers with the decagon being the central polygonal in the numeric chain. As with the examples illustrated in Fig. 371, this eccentric method of providing for atypical and unexpected regions of local symmetry within an otherwise isometric structure offers tremendous potential for creating new patterns to contemporary geometric artists. As mentioned in reference to Fig. 375a above, these polygonal arrangements can be used as layouts for radii matrices that can then be used to create very successful underlying tessellations with primary n-fold polygons separated by a connective matrix of irregular pentagons, barrel hexagons, and other case-sensitive polygons. By way of example, Fig. 376 further develops the arrangement of hendecagons, decagons, and nonagons from Fig. 375h into an underlying tessellation suitable for creating a variety of patterns. Figure 376a illustrates the isometric grid along with its dual hexagonal grid. Each included angle within the hexagonal grid can be represented as a 1/3 division of the circle. In Fig. 376b the included angles of the two varieties of hexagon now correspond to alternating 10- and 11-fold divisions of the circle. It is important to note that the 10- and 11-fold radii do not actually align, and that the edges of both varieties of hexagon are made up of two noncollinear lines that intersect at the small black dots. What appear to be hexagons are, therefore, actually dodecagons. The nonaligning of the edges is a standard feature of this variety of geometric structure, and all of the examples from Fig. 375 (with the exception of C) have this anomaly. From the perspective of geometric precision this might be regarded as a fault, but for the purposes of creating geometric designs this anomaly is of no consequence. Figure 376c places 10 and 11 radii at the vertices of this (pseudo) hexagonal grid, and 9 radii at the center of each beige (pseudo) hexagon. Again, the black dots indicate the intersections of the noncollinear radii. From this radii matrix the underlying tessellation in Fig. 376d is easily created using the standard conventions for creating such tessellations. An important feature of the matrix of pentagons and barrel hexagons that separate the primary polygons is that their being located at the points of intersection of the not-quite-collinear radii overcomes any problems associated with the nonaligned radii, thereby allowing for the primary polygons (in this case hendecagons, decagons, and nonagons) to be regular. Figure 377a illustrates the obtuse pattern (by author) created from the underlying tessellation created in the previous figure. Figure 377b demonstrates how this same pattern can be created from the nesting of primary polygons as per Fig. 375h. As explained, the radii associated with these primary polygons are not quite aligned, and therefore the polygonal edges are not able to be in precise edge-to-edge contact. This produces difficulties in laying out the pattern lines. By contrast, the matrix of pentagons and barrel hexagons in Fig. 377a allows for the regularity of the primary polygons to not be in conflict with one another; making the application of the pattern lines less problematic. Figure 377c demonstrates the dual relationship between these two underlying tessellations. This is analogous to many other examples, especially in the fivefold system [Fig. 200]. Figure 377d shows a very acceptable interweaving version of the obtuse design that can be created from either of these dual underlying tessellations. Figure 378 demonstrates the derivation of the acute pattern (by author) that is produced from the underlying tessellation in Fig. 376d. This design combines 9-, 10-, and 11-pointed stars in a balanced arrangement that is fully in keeping with the aesthetics of this ornamental tradition. Just as decagons tessellate on their own (with concave hexagonal interstice regions as per Fig. 200a), so also will decagons successfully tessellate with their numerically adjacent nonagons and hendecagons. This is an expression of the principle of adjacent numbers. The median and two-point patterns that this underlying tessellation creates are also very acceptable (not shown).

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Fig. 378

3.2.4 Orthogonal Designs with Multiple Regions of Local Symmetry

The most common variety of orthogonal pattern with multiple regions of local symmetry employs 12-pointed stars at the vertices of the orthogonal grid and 8-pointed stars at the center of each square repeat unit. Several underlying tessellations were used that produce such patterns, but the most common surrounds each underlying octagon with a ring of pentagons, and separates each dodecagon with a barrel hexagon. The simplicity of constructing this underlying tessellation from a radii matrix is demonstrated in Fig. 304a. Evidence for the use of this radii matrix in producing the median pattern created from this underlying tessellation is found in the Topkapi Scroll.Footnote 62 Patterns that can be created from this underlying tessellation can also be made from a tessellation of edge-to-edge dodecagons and octagons, with concave hexagonal interstice regions. As with other examples of a single design that can be created from either of two generative tessellations, these underlying tessellations have a dual-like relationship. Figures 379–382 illustrate the derivation of patterns from each of the four pattern families from both of these two closely related underlying tessellations. The three acute patterns with 8- and 12-pointed stars in Fig. 379 demonstrate how subtle differences in the angles of the applied pattern lines can change the overall character of what is otherwise the same design. The Mengujekid example in Fig. 379b is from the Kale mosque in Divrigi, Turkey (1180-81), and has wider interweaving lines and slightly more acute angles in the five-pointed stars. This example also uses a truncated pattern line treatment within the underlying barrel hexagons. All of these features produce a visual quality that is distinct from the version of this acute pattern in Fig. 379e. Many examples of this acute design are found throughout the Islamic world, although the specific proportions will vary from example to example. In addition to line thickness, these variations are caused by subtle differences in the angles of the crossing acute pattern lines. Notable examples of this acute design include a Seljuk border pattern in the entry of the Friday Mosque at Gonabad (1212) [Photograph 23], and two late Abbasid exemplars: one from the Palace of the Qal’a in Baghdad (c. 1220), and the other from the Mustansariyya in Baghdad (1227-34). A Mamluk example of this design was used as a side panel of the minbar of the Amir Azbak al-Yusufi complex in Cairo (1494-95) [Photograph 46]. The slightly less acute angles of the five-pointed stars combined with narrower widened lines creates a more open design with background elements that are proportionally larger than those of Fig. 379b. A tiling version of this design was used within a recessed arch tympanum in the southern corner of the southeast iwan in the Friday Mosque at Isfahan [Photograph 29]. This is stylistically similar to several of the Seljuk designs in the nearby northeast dome chamber (1088-89), and its provenance appears to be Seljuk from the late eleventh or early twelfth century. The pattern lines of the 12-pointed stars of Fig. 379h are comprised of 12 parallel sets. This pulls the crossing pattern lines inward from the midpoints of the underlying dodecagon in Fig. 379i. This was a common convention among artists in the Maghreb and North Africa, and the Ottoman example illustrated here comes from the Great Mosque of Sfax in Tunisia (eighteenth century). An earlier example from the Seljuk Sultanate of Rum was used in the minaret of the Great Mosque of Siirt in Turkey (1129). While both underlying tessellations in Fig. 379 will make all three variations of the acute design, the tessellation with the ring of pentagons and barrel hexagons provide contact points for all of the key pattern line placements. By contrast, the 8- and 12-pointed star rosettes within the alternative tessellation of edge-to-edge octagons and dodecagons require an additive process that is not directly the product of the underlying generative tessellation. Figure 380 demonstrates the construction of two median patterns from these two underlying tessellations. Figure 380b shows the standard median derivation without design modification. An Ayyubid carved stone example of this design is found in the city walls of the Bab Antakeya in Aleppo (1245-47); an Ilkhanid example is found in the portal of the Gunbad-i Gaffariyya in Maragha, Iran (1328); and Timurid examples are found in the cut-tile mosaics at both the Bibi Khanum in Samarkand (1398-1404), and the Friday Mosque at Herat (fifteenth century). The design in Fig. 380e was employed in the triangular side panels of the minbar at the Sultan Mu'ayyad mosque in Cairo (1415-21). This example uses the arbitrary modification that was typically applied to median patterns [Fig. 223] and was especially popular among Mamluk artists. Another feature of this design is the arbitrary incorporation of heptagonal elements within the pattern matrix. Although occasionally found in the geometric patterns of other Muslim cultures, for example those produced during the Seljuk Sultanate of Rum, arbitrary inclusions such as these heptagons are a relatively common feature within the Mamluk geometric tradition. As with the acute patterns created from the underlying tessellation of edge-to-edge dodecagons and octagons, not all of the crossing pattern lines of the median 12-pointed stars are fixed upon the midpoints of the dodecagonal and octagonal edges in Fig. 380a and d. For this reason, the alternative underlying tessellation that includes pentagons and barrel hexagons is generally more convenient for producing these median designs. Figure 381 illustrates two obtuse patterns that can be created from the same two underlying tessellations. Unlike the previous examples, all of the pattern lines of the 8- and 12-pointed stars in the obtuse example in Fig. 381a are directly associated with the midpoints of the underlying dodecagonal edges. For this reason each of the two underlying tessellations is equally expedient for generating these obtuse designs. Mamluk examples of the design in Fig. 381b are found in the blind arches that surround the exterior drum of the dome at the Hasan Sadaqah mausoleum in Cairo (1315-21), and in a window grill on the drum of the dome at the contemporaneous Amir Sanqur al-Sa’di funerary complex in Cairo (1315). Ilkhanid examples from roughly the same period include a vaulted ceiling panel from the mausoleum of Uljaytu in Sultaniya, Iran (1307-13), and a cut-tile mosaic border at the Gunbad-i Gaffariyya in Maragha, Iran (1328). Later eastern examples include a Qarjar cut-tile mosaic arch over the reconstructed entry door at the Aramgah-i Ni’mat Allah Vali in Mahan, Iran (nineteenth century); a cut-tile mosaic panel from the Qarjar restorations of the Malik mosque in Kerman, Iran; and a stone mosaic panel created by Mughal artists at the tomb of Akbar in Sikandra, India (1612). The obtuse design in Fig. 381e is also from the mausoleum of Uljaytu. This example includes a modification to the decagonal region that disguises the 12-pointed star through an arbitrarily added sixfold motif that is similar in concept to the well-known fivefold modification [Fig. 224a]. Figure 382b shows a Mamluk two-point pattern also produced from either of these underlying tessellations, but the example with the underlying pentagons and barrel hexagons has more contact points with the crossing pattern lines. As with the median pattern in Fig. 380e, this design incorporates the same variety of modification to the 12-pointed stars [Fig. 223]. This example is from the triangular side panels of the minbar at the Princess Asal Bay mosque in Fayyum, Egypt (1497-99).

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As with the previous examples, the two-point pattern in Fig. 383 also places 12-pointed stars at the vertices of the orthogonal grid and 8-pointed stars at the center of each square repeat unit. However, the underlying tessellation for this design separates the dodecagons and octagons with a barrel hexagon rather than the two pentagons from the previous example. This underlying tessellation also contains the characteristic cluster of six pentagons surrounding a thin rhombus that is a frequent feature of fivefold underlying tessellations. As detailed previously, this cluster of six pentagons is well suited to producing obtuse and two-point patterns. However, for acute and median patterns, the pentagons require truncation for the creation of well-composed patterns. This two-point pattern is of Mamluk origin, and comes from the Sultan Qaytbay funerary complex in Cairo (1472-74). Figure 304b shows how the underlying tessellation that produces this two-point pattern is constructed from the same radii matrix as the previous set of patterns with 8- and 12-pointed stars.

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Fig. 383

The two median designs in Fig. 384 also place 12-pointed stars at the vertices of the orthogonal grid and 8-pointed stars at the center of the square repeat unit. The underlying tessellations responsible for these two designs are similar to that of Fig. 304a, except for the ring of eight rhombi replacing the ring of eight pentagons. The proportions of the eight-pointed stars in Fig. 384a are determined by mirroring the lines at the edges of the underlying octagons, whereas the eight-pointed stars in Fig. 384b are the product of the pattern lines within the rhombi continuing toward the center of the repeat until they meet with other continued lines. The example in Fig. 384a is In’juid from the tympanum in the east portal of the Friday Mosque at Shiraz (1351), and the design in Fig. 384b is Muzaffarid from the Friday Mosque at Yazd (1365). The relative closeness in time and proximity between these two very similar median designs suggests the possibility of their being produced by the same artist or atelier. The design in Fig. 384a was also used in the Timurid cut-tile ornament of the Gur-i Amir complex in Samarkand (1403-04). The arrangement of underlying polygons that surround the region with eightfold local symmetry in the two designs in this figure are analogous to a type of pattern created from the fivefold system [Figs. 236, 237, and 253]. This employs an underlying arrangement of pentagons and/or wide rhombi that surround the primary centers of local symmetry such that their vertices are aligned with the primary radii rather than the midpoints of their edges. This leaves either an n-fold interstice star at the primary centers of local symmetry, as per Fig. 384b, or a ring of triangles surrounding the n-fold primary polygon, as per Fig. 384a. Most of the designs with this variety of polygonal arrangement, be it fivefold or otherwise, are from the eastern regions, and date to the period after the Mongol conquest. Figure 385 demonstrates a construction for the underlying tessellation that produces these designs. Step 1 starts with the radii matrix from Fig. 304a. However, rather than the central region being filled with the ring of pentagons surrounding an octagon, this step shows just the dodecagons and barrel hexagons. Step 2 simply mirrors the indicated angles to produce the ring of rhombi; and Step 3 places an octagon at the center of the repeat unit, thus completing the tessellation.

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Figure 386 illustrates another median pattern with the same placement of the 8- and 12-pointed stars. As with the previous example, this also employs the unusual arrangement of vertex-to-vertex pentagons that surround both the 8-fold and 12-fold centers of radial symmetry. This produces the underlying triangular regions that surround each primary polygon, and the pattern line application along the edges of the primary polygons employs two points rather than the singular midpoints that are typical of median patterns. The cluster of four contiguous rhombi produces a design feature that is also rather unusual, and occasionally found in some fivefold patterns. This nonsystematic Timurid design is from the Ulugh Beg madrasa in Samarkand (1417-20) [Photograph 88], and it is interesting to note that one of the analogous fivefold designs is also located at this madrasa [Fig. 236]. These are presumably the work of the same artist. Figure 387 demonstrates a simple construction of the underlying generative tessellation using the radii matrix technique. The radii matrix places 24 radii at each corner of the square repeat unit, and 16 radii at the center. Step 1 draws a circle at the indicated intersection of the red radii that is tangent to all three of the blue radii. A pentagon is drawn inside the circle with vertices at the intersect points of the red radii and circle, as well as the points at which the circle is tangent with the blue radii. Step 2 mirrors this pentagon. Step 3 mirrors a single edge of each pentagon; establishing one quarter of a 12-pointed star. Step 4 mirrors the lines from Step 3 to create the rhombi. Step 5 rotates these elements throughout the square repeat. And Step 6 shows the completed tessellation.

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Fig. 387

Figure 388 shows two alternative constructions for an acute design that places 16-pointed stars on the vertices of the orthogonal grid and 8-pointed stars at the center of each square repeat unit. The underlying tessellation in Fig. 388a is comprised of edge-to-edge 16-gons and octagons, with elongated concave octagonal interstice regions separating the adjacent 16-gons. The pattern lines associated with the two adjacent parallel edges of the 16-gons overlap to create a small rhombus at the center of the interstice region. This is an atypical feature that is not generally in keeping with this ornamental tradition. The historical example of this design changes the angles of the pattern lines associated with these two edges of the underlying 16-gon such that they create the more familiar dart shape shown in Fig. 388b. This results in the new dart shapes being larger and differently proportioned than the adjacent darts that surround the 16-pointed stars. Figures 388d and e illustrate an alternative method of creating this design that uses an underlying tessellation with the commonly found feature of a ring of pentagons. The example in Fig. 388d shows a reasonably successful acute pattern, although the proportion of the five-pointed stars created from the large pentagons (yellow) is not ideal. This less desirable feature is resolved in Fig. 388e through the modification of the 16-pointed star rosette using the common Mamluk method [Fig. 223]. The end result of this modification is very successful, and the Mamluk design pictured in this figure is from the Sultan al-Mu'ayyad Shaykh complex in Cairo (1412-22) [Photograph 60]. Figure 389 demonstrates two additional designs produced from the same underlying tessellation with 16-gons and octagons as the previous example. Figure 389a shows an acute pattern with the shared points of the 5- and 16-pointed stars being moved off of the midpoints of the 16-gon inward toward the vertices of the square repeat unit. This elongates the five-pointed stars, changing the overall aesthetic effect in a manner that is most frequently found in the Maghreb. This example was used in a Mudéjar window grille in the ibn Shushen Synagogue of Toledo (1180), and now known as the Santa Maria la Blanca. Another example of this design was used for an illuminated frontispiece of the Quran (1310) commissioned by the Ilkhanid Sultan Uljaytu.Footnote 63 Figure 389b shows a Timurid median pattern created from the same underlying tessellation that is found in the entry iwan of the Ulugh Beg madrasa in Samarkand (1417-20) [Photograph 89]. Figure 390 illustrates a construction of the underlying tessellation from Fig. 389. The radii matrix places 32 radii at each corner of the square repeat unit, and 16 radii at the center. Step 1 shows the standard placement of a circle at the intersection of the chosen radii that is tangent with the red radii, and lines that are perpendicular to the blue radii that intersect with the circle and blue radii. This establishes the edge of the 16-gon, the edge of the octagons, and the separating pentagon. Step 2 shows all three of these features. Step 3 mirrors the pentagon. Step 4 rotates these features throughout the square repeat, identifying the second variety of pentagon in the process. These are located at the midpoints of the repetitive edges. Step 5 shows the completed tessellation. Step 6 shows a modification to the ring of eight pentagons that produces the design in Fig. 391. This modification transforms the tessellation, and the derived patterns, from having 8- and 12-fold local symmetries to 4- and 12-fold symmetries, thereby eliminating the eight-pointed stars from the finished pattern. This was used for a Mamluk acute design from the Sultan Hasan funerary complex in Cairo (1356-63). The application of crossing pattern lines to the central underlying squares foregoes the more acute angles of the acute family, and thereby provides for the inclusion of the regular octagon in this location.

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The acute pattern in Fig. 392d is comprised of 16-pointed stars at the vertices of the orthogonal grid and 12-pointed stars at the centers of each repeat unit. This design is a modification of the standard acute pattern shown in Fig. 392b that follows the common convention, especially among Mamluk artists, for changing acute and median patterns [Fig. 223]. Applying this modification to both varieties of primary star radically changes the overall appearance of the design. This modified pattern was used in the bronze entry doors that first graced the Sultan al-Nasir Hasan funerary complex in Cairo (1356-63), but were moved to the Sultan al-Mu’ayyad complex in 1416-17, where they remain to this day. The underlying tessellation in Fig. 393 employs a modification to the tessellation in Fig. 392a wherein the six pentagons that surround each thin rhombus are truncated into trapezoids. This follows the convention for modifying star rosettes of the acute family that was established within the fivefold system [Fig. 198]. This modification changes what would otherwise be 6 five-pointed stars into 6 dart motifs. The earliest known example of this acute pattern was produced by Seljuk artists for a panel within the muqarnas hood of the mihrab in the Friday Mosque at Barsian (1105). This is a surprisingly early date for such a sophisticated design. Rather than using widened or interweaving lines (as per the illustration), the tiled expression of this example from Barsian follows the Seljuk geometric aesthetic exemplified by the roughly contemporaneous geometric designs in the nearby northeast dome chamber of the Friday Mosque at Isfahan (1088-89). The interweaving expression pictured in Fig. 393 was used by Mamluk artists in several locations, including: a window grill in the mosque of Altinbugha al-Maridani in Cairo (1337-39) [Photograph 56]; a curvilinear variation from the bronze entry doors of the Sultan al-Nasir Hasan funerary complex in Cairo (1356-63); and the triangular side panel of the wooden minbar (1468-96) commissioned by Sultan Qaytbay, and currently on display at the Victoria and Albert Museum in London. Figure 394 demonstrates a construction of the underlying tessellation with 16-gons and dodecagons from Figs. 392 and 393. This makes use of a radii matrix with 32 radii placed at the corners of the square repeat unit, and 24 radii at the centers. Step 1 places a circle at the intersection of two blue radii that is tangent to the red radii. Lines have been drawn that are perpendicular to the blue radii and intersect with the circle and blue radii. Step 2 uses these lines to produce the 16-gon and dodecagon, as well as the separating pentagon. Step 3 mirrors the pentagon as well as draws two additional pentagons that connect the end points of projected red and blue radii. Step 4 creates a third set of pentagons along the repetitive edges by mirroring lines from Step 3. Step 5 rotates these elements throughout the square repeat. And Step 6 shows the completed tessellation, along with the potential truncated cluster of six pentagons for the design from Fig. 393. The two-point pattern in Fig. 395 also employs 16-pointed stars at the corners of the square repeat unit and 12-pointed stars at the center of each repeat. This design was used in several Mamluk locations, including the triangular side panel of the stone minbar at the Sultan al-Zahir Barquq in Cairo (1384-86) [Photograph 57]; the triangular side panel of the wooden minbar at the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81); and the triangular side panel of the wooden minbar at the Sultan Qansuh al-Ghuri complex in Cairo (1503-05). The underlying tessellation separates the 16-gons and dodecagons with a barrel hexagon, and separates the adjacent 16-gons with an arrangement of ten pentagons surrounding two mirrored irregular heptagons. The precise placement of the pattern lines of the two-point family is inherently flexible, and in this case those associated with the heptagons have been carefully placed so that they produce twin seven-pointed stars that have true sevenfold rotational symmetry. Figure 396 illustrates two acute patterns that are also created from the underlying tessellation in Fig. 395. The standard acute pattern (by author) that is created from this underlying tessellation is represented in Fig. 396a, but there are no known examples of this design in the historical record. The design in Fig. 396b is a Mamluk example that modifies the 12- and 16-pointed stars using the technique demonstrated in Fig. 223 wherein the surrounding five-pointed stars are transformed into darts. The pattern lines associated with the barrel hexagons in Fig. 396b have been arbitrarily adjusted to produce two nearly regular heptagons within the pattern matrix. This form of pattern adjustment was frequently employed by Mamluk artists, and was occasionally used elsewhere. This Mamluk acute design is from a carved stone panel at the base of the minaret at the Mughulbay Taz mosque in Cairo (1466). Figure 397 demonstrates how the construction of the underlying tessellation used in creating the patterns in Figs. 395 and 396 employs the same radii matrix as that of Fig. 394. This is made up of 32 radii placed at the corners, and 24 radii at the centers of the square repeat unit. However, the underlying tessellation shown in Fig. 397 employs an entirely different derivation from this radii matrix. In creating the underlying tessellation, Step 1 draws an almost regular heptagon at the midpoint of the edge of the square repeat. The size is determined by the vertex of a regular heptagon intersecting the blue radii of 32-fold symmetry, as per the detail. The detail also shows how two of the heptagonal vertices do not quite fall upon the vertices of the pair of intersecting red radii. These nonaligned heptagonal vertices are therefore moved so that they rest on the intersection of the red radii, making the heptagon slightly irregular. Step 2 draws two circles; one centered on a red radius of 32-fold symmetry and they other on a red radius of 24-fold symmetry. These circles are tangent to the blue radii. Lines that are perpendicular to these red radii are drawn that determine the size of the 16-gon and dodecagon. Step 3 shows these primary polygons as well as the two pentagons and a barrel hexagon between the 16-gon and dodecagon. Step 4 mirrors these elements and introduces pentagons along the edges of the repeat that are determined by mirroring the edge of the heptagon. Step 5 rotates these elements throughout the square repeat. And Step 6 shows the completed tessellation.

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It is rare for orthogonal patterns with just two primary star forms to not locate these on the vertices and center of the square repeat unit. Figure 398 shows an unusual acute design that places 16-pointed stars at the vertices of the orthogonal grid and four 13-pointed stars within the field of each square repeat unit. Patterns that place primary star forms within the field of the repeat ordinarily utilize the more typical locations first; for example, the vertices of the orthogonal grid and orthogonal dual grid, and the midpoints of the edges of each repeat unit. It is interesting to note that the centers of the 13-pointed stars appear to fall upon the vertices of the semi-regular 4.82 tessellation of squares and octagons [Fig. 89]. However, close examination reveals that there are two different distances between the locations of the 13-pointed stars, thereby disqualifying this from adhering to the 4.82 semi-regular symmetry. This interesting design is from the Topkapi Scroll, but is not otherwise known to the historical record.Footnote 64 A construction of the underlying tessellation for this eccentric acute pattern is demonstrated in Fig. 399. Step 1 places 32 radii at each vertex of the square repeat unit. The four pairs of emphasized radii (black) have internal angles of 135°. This angle is close to the 138.4615…° found within a 13-fold division of a circle, indicating that a 13-pointed star can be placed at these locations. Step 2 introduces 26 radii at these four locations. The small black dots indicate the intersection points for the 32- and 26-fold radii that do not quite align. Therefore, the red radii connecting these centers are not quite collinear, although they appear so. Step 3 determines the edges of the tridecagons, as well as the separating pentagon using the standard formula. Step 4 draws the two completed tridecagons as well as the two separating pentagons. Step 5 uses intersect points within the radii matrix to produce further pentagons. This step also determines the edge of the 16-gon using a circle in tangent with the blue radii. Step 6 mirrors these elements. Step 7 completes one quadrant by making the final variety of pentagon along the edges of the repeat. Step 8 rotates these elements throughout the repeat. And Step 9 completes this rather remarkable tessellation. It is worth noting that the un-inked scribed “dead lines” of this drawing in the Topkapi Scroll show both the radii matrix and the underlying tessellation as the generative schema employed in the design of this pattern.

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Fig. 398

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Fig. 399

There are relatively few historical orthogonal patterns that employ three primary star forms within their overall make up. The acute pattern in Fig. 400 is one of the most complex orthogonal designs found within Islamic geometric art. This is comprised of 12-pointed stars placed at the vertices of the orthogonal grid, octagons at the centers of each repeat unit, 10-pointed stars at the midpoints of the repetitive edges, and 9-pointed stars within the field of the repeat. The nine-pointed stars rest upon the diagonals of the square repeat. As with so many of the particularly complex Islamic geometric designs, this is the product of artists working during the Seljuk Sultanate of Rum, and is found at several Anatolian locations, including the Kayseri hospital (1205-06); the Agzikara Han near Akseri (1236-46); and the Çifte Kumbet in Kaysari (1247). Figure 401 shows a construction for the underlying tessellation that creates this pattern. Once again, this begins with the creation of a radii matrix. Step 1 places 24 radii next to 20 radii such that they share a horizontal radius (red). The 45° diagonal radius of the 24 radii is extended to meet the extended vertical radius of the 20 radii to establish the fundamental domain (blue) of the eventual design. The 81° angle of this construction is close to the 80° angle of a ninefold division of the circle, indicating the potential location of a nine-pointed star. Step 2 places 18 radii at this location. This is aligned with the 45° diagonal radius from the 24 radii. Note: Detail 1 shows how the relevant radius from these 18 radii does not connect with the center of the 20 radii, and is not quite parallel with the closest radius from the 20 radii. Step 3 begins the process of correcting this situation by copying the 45° diagonal radius from the 24 radii to a location midway between the 18 and 20 radii. As shown in the Detail 2, Step 4 trims the two nonaligned radii with this copied diagonal line. Step 5 moves the group of 18 radii along the copied diagonal line so that the two trimmed radii meet. This intersection is indicated with the black dot. While these two radii are not actually collinear, for the purposes of this design process they function as thought they are. Step 6 introduced 8 radii at the 45° upper corner of the fundamental domain. Step 7 determines the edges of the three primary polygons, as well as the two separating pentagons. Step 8 draws the dodecagon, decagons, and nonagon, as well as the connecting pentagons and barrel hexagon. Step 9 mirrors these elements and the fundamental domain. Step 10 rotates the elements from Step 9 four times, and Step 11 shows the completed tessellation.

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Fig. 400

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Fig. 401

The acute design in Fig. 402 is from one of the side panels of the minbar at the Sultan Qaytbay funerary complex in Cairo (1472-74). This Mamluk example places 16-pointed stars at the vertices of the orthogonal grid, 12-pointed stars at the center of each square repeat unit, and 10-pointed stars at the midpoints of each edge of the repeat. In keeping with Mamluk aesthetic practices, the pattern lines within the concave hexagons have been adjusted to produce twin heptagons. With its three regions of local symmetry, this is one of the more complex orthogonal geometric patterns created by Mamluk artists. Figure 403 shows a construction for the underlying tessellation used for creating the pattern in Fig. 402. Step 1 illustrates a radii matrix with 32 radii at the corners of the square repeat, 24 radii at the center of the repeat, and 20 radii at the midpoint of the edges of the repeat. Step 2 determines the edges of the primary polygons, as well as two of the pentagons using the standard formula of a circle in tangent with the red radii and applied perpendicular lines. Step 3 draws the 16-gon, dodecagons and decagon, as well as the pentagons and barrel hexagon. These pentagons and barrel hexagon are determined from the intersections of the primary polygonal edges and once the primary polygons have been established, are implicit within the radii matrix. Step 4 mirrors these elements, and adds two pentagons to complete one quadrant. The large circle provides for regularity in the edge lengths of the new pentagons. Step 5 rotates these elements throughout the square repeat. And Step 6 completes the tessellation. Note: the truncation of the clustered six pentagons, and resulting concave hexagons is also indicated. This variety of modification [Fig. 198] is required for the acute pattern in Fig. 402.

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Fig. 402

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Fig. 403

The design in Fig. 404b places 16-pointed stars at the vertices of the orthogonal grid, 8-pointed stars at the center of each square repeat unit, 12-pointed stars at the midpoints of each repeat unit, and 10-pointed stars on the repetitive diagonals within the field. This acute pattern is from the Kemaliya madrasa in Konya (1249), and shares its regions of local symmetry with two other examples from the Seljuk Sultanate of Rum; although these others are created from the extended parallel radii design methodology rather than the polygonal technique [Fig. 81]. Each of the three primary star forms in this example is comprised of sets of parallel lines, all of equal width. This is a distinctive feature more commonly seen in the geometric art of the Maghreb. As in other examples that have this feature, the crossing pattern lines of the primary stars will not necessarily fall upon the midpoints of the primary polygonal edges, but will be moved inward toward the polygonal centers. This allows for the uniformity in the width of the parallel pattern lines that make up the primary stars. As shown in the upper left panel in Fig. 404a, the width of the parallel lines are established within the decagonal region, and copied into the 16-gons and dodecagons. The eight-pointed stars are atypical in that they are not produced directly from an underlying octagon, but through extending the lines within the ten-pointed stars toward the center of the square repeat unit, and rotating these four sets of diagonal parallel pattern lines by 45°. Figure 405 illustrates a construction for the underlying tessellation used for making the design from Fig. 404. Step 1 shows one quadrant of a radii matrix with 32 radii in the corner of the repeat unit (upper left), and 24 radii at the midpoints of the repeat (upper right and lower left). The two black radii have an included angle of 150° which is relatively close to the 144° found in a tenfold division of a circle. Step 2 introduces 20 radii at this point. The radii between the 20- and 24-fold centers are not aligned. To correct this, the black radius between the 32- and 20-fold centers is copied to points that are midway between the two 20- and 24-fold centers. Step 3 trims the radii with these copied lines. Step 4 moves the 20 radii along the black diagonal lines until the trimmed radii intersect. The black dots indicate the intersections of these radii, and what functions as a line of radius between the 24- and 20-fold centers is actually two intersecting noncollinear radii. Step 5 determines the edges of the primary polygons as well as the separating pentagons. Step 6 draws the 16-gon, dodecagon and decagons. Step 7 mirrors the dodecagon and fills in the separating matrix of pentagons. Step 8 finishes the quadrant, and Step 9 shows the completed tessellation in a full repeat. Note: The four large pentagons at the center of the repeat are out of scale to the other pentagonal elements. This would ordinarily cause problems. However, by applying pattern lines to these regions as illustrated in Fig. 404, this problem is elegantly overcome through the introduction of the eight-pointed star.

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Fig. 404

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Fig. 405

Figure 406 demonstrates a highly versatile method of creating orthogonal patterns with multiple regions of local symmetry. This repetitive schema is an orthogonal corollary to the isometric method illustrated in Figs. 371 through 374. However, whereas the isometric methodology has no historical precedent, several patterns that conform to this orthogonal repetitive structure are known historically. The construction of tessellations with this technique requires a central polygon with sides that are multiples of four. Four n-sided polygons are placed edge-to-edge in rotation around the central polygon, and these are divided in half to determine the square repetitive unit. This unit is reflected on its vertical and horizontal edges to create the larger square repeat unit with translation symmetry. Unless an interweaving line introduces chirality to the pattern lines, this variety of pattern always conforms to the cmm plane symmetry group. The shaded squares in Fig. 406 indicate the oscillating square feature of this variety of design. In this regard, they conform to the historical examples illustrated in Figs. 23 through 25. However, the application of the polygons, and secondary infill of further polygons provides for greater complexity than most historical oscillating square patterns. Figure 406a uses a central octagon surrounded by four heptagons. Figure 406b adds nonagons to the central octagon, and Fig. 406c uses four hendecagons (11-gons) surrounding the central octagon. Figure 406d places heptagons around the central dodecagon. Figure 406e places four octagons around the central dodecagon. Figure 406f shows four nonagons around the central dodecagon. Figure 406g adds four dodecagons around the central dodecagon. And Fig. 406h places four tridecagons (13-gons) around the central dodecagon. Each of these has been further developed with the subdivision of the interstice regions into secondary polygons, and each is well suited to producing very acceptable geometric designs. Figure 407 illustrates the design potential of using just the central polygon and the four surrounding polygons for creating patterns. In these two examples the interstice region is passive, and has not been filled with secondary polygons. With the exception of the secondary polygons, the underlying tessellation in Fig. 407a is the same as that of Fig. 406a, and that of Fig. 407c is the same as Fig. 406c. The median pattern in Fig. 407b is comprised of octagons and seven-pointed stars. This was used historically in many locations, and is one of the better-known patterns with seven-pointed stars, although clearly not created from the sevenfold system. The application of the octagon within the underlying octagon is unusual, and certainly an eight-pointed star could have been used in this location. The proportions of the octagon are determined by using two points on every other underlying octagonal edge. In this case, the two points are determined by dividing the edge into quarters. Historical examples of this design include: a Jalayirid arch spandrel at the Mirjaniyya madrasa in Baghdad (1357), and a small carved stone relief panel in the Mamluk iwan of the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81). The pattern in Fig. 407d (by author) uses the same design process of only using the primary polygons, with the interstice regions being passive, and favoring an octagon within the underlying octagons over an eight-pointed star. The main difference between these two designs is that the historical example combines 7-pointed stars with octagons and the other uses 11-pointed stars and octagons. The two designs in Fig. 408 (by author) demonstrate the efficacy of using secondary polygonal elements that fill the interstice regions between the two primary polygons. The acute pattern in Fig. 408b is comprised of 8- and 12-pointed stars and is created from the underlying tessellation of central dodecagons and surrounding octagons. The placement of the pattern lines allow for the incorporation of octagons within the pattern matrix. The underlying tessellation is the same as Fig. 406e. The obtuse pattern in Fig. 408d is made up of 9- and 12-pointed stars. The underlying tessellation in Fig. 408c, and the resulting obtuse pattern, are analogous to examples created from the fivefold system [Fig. 235b].

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Fig. 406

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Fig. 407

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Fig. 408

Just as in with the analogous isometric examples, Fig. 409 demonstrates how the orthogonal arrangement of a central polygon with edges that are a multiple of four, and four surrounding n-sided polygons can be used to make radii matrices from which very acceptable underlying tessellations can be created. A radii matrix has been added to the tessellation of dodecagons and nonagons in the upper left panel of Fig. 409a. The radii can be seen to converge on a point that allows for another radial center with 20-fold symmetry. This indicates that a ten-pointed star can be placed at these locations. The upper right panel shows just the radii matrix. The lower left panel of Fig. 409a illustrates a tessellation of primary dodecagons, decagons and nonagons that are separated by a polygonal matrix of pentagons and barrel hexagons. And the lower right panel shows just the tessellation and repetitive structure. Note: the decagons placed at the two adjacent centers with 20-fold symmetry are conjoined, and this underlying feature can produce attractive patterns [Fig. 192]. The acute pattern with 9-, 10-, and 12-pointed stars in Fig. 409b (by author) is very acceptable to the aesthetics of this ornamental tradition.

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Fig. 409

Figure 410 illustrates another approach to producing underlying tessellations with the same repetitive schema as the examples in Figs. 406 through 409. This technique dispenses with the use of the primary polygons as the first step in establishing a radii matrix (as per Fig. 409), and starts directly with the production of the radii matrix. Similarly, rather than focusing on the reflected square as the repetitive element during the design process, this technique begins with the concave octagonal shield shape that is always associated with this type of repetitive structures. As indicated in the examples from Figs. 406 through 409, the shield-shaped repetitive cells require 90° rotation to cover the plane. The shield shapes can be thought of as a square with 90° corners that have been rotated to accommodate symmetries with n-fold symmetry at the midpoints of the otherwise square. The upper left of Fig. 410 is a square with 16 radii at each corner, and 16 radii at each midpoint of the square’s edges. Step 1 demonstrates how the midpoints can accommodate a different n-fold radial center by rotating each corner an amount that conforms to the introduced symmetry—in this case 18-fold. The radii at the four corners of the square must always be a multiple of four to maintain the right angle, but the introduced n-fold symmetry at the erstwhile midpoints is entirely flexible. Step 2 extends the radii of the 16- and 18-fold centers until they intersect, thus completing the radii matrix. Step 3 determines the size of the octagon and nonagon in the standard method using circles and perpendicular lines. Step 4 draws the primary polygons, as well as the separating barrel hexagon and pentagons. Step 5 mirrors these elements around the periphery of the shield shape. Step 6 mirrors the pentagons to the other edges of each nonagon. Step 7 fills the remaining space with additional pentagons, triangles, and a central barrel hexagon. Step 8 shows the completed tessellation. Figure 411 illustrates four patterns (by author) that are made from the underlying tessellation from Fig. 410. Each is a combination of eight- and nine-pointed stars. The dashed red lines indicate reflection symmetry, and the black dots indicate points of rotational symmetry. The dashed black lines represent the shield shapes in fourfold rotation around each rotation point. As mentioned, these designs have the same repetitive structure as the design from Figs. 406 through 409. Figure 411a shows an acute pattern. Figure 411b shows an obtuse pattern. Figure 411c shows a median pattern, and Fig. 411d shows a two-point pattern.

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Fig. 410

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Fig. 411

These two approaches to creating complex orthogonal geometric patterns with multiple regions of local symmetry that conform to the aesthetic standards of this artistic tradition offer tremendous scope for contemporary artists and designers working in this discipline.

3.2.5 Rectangular Designs with Multiple Regions of Local Symmetry

Nonsystematic designs with rectangular repeat units range from the rather simple to the very complex. The application of multiple regions of local symmetry follows several conventions that are specific to this repetitive schema. The n-fold symmetry located at the corners of the rectangular repeats unit is, perforce, divisible by two. Similarly, if the center of the rectangular repeat unit is also populated with a region of local symmetry, this will likewise be divisible by 2—the center being a vertex of the rectangular dual grid. If there are primary stars located upon the edges of the repeat unit, these can have either even or odd number of points, as can primary stars located within the field of the repeat unit.

Figure 412 is an acute field design that, while relatively simple in its geometric composition, is nonetheless appealing to the eye. This places octagons at the vertices of a rectangular grid, as well as at the vertices of the identical dual grid, and the geometric information contained within the repeat unit is identical to that of its dual. The underlying tessellation is comprised of just two polygonal elements: irregular pentagons and irregular hexagons, and the pattern adheres to the cmm plane symmetry group. This is a Mamluk design from an incised stone border in the entry portal of the Sultan al-Nasir Hasan funerary complex in Cairo (1356-63) [Photograph 58].

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Fig. 412

The design in Fig. 413 is also from the entry portal of the Sultan al-Nasir Hasan funerary complex in Cairo [Photograph 58]. This pattern places eight-pointed stars at the corners of a rectangular repeat unit and regular hexagons at the midpoints of each repetitive edge. This also places an eight-pointed star at the center of the repeat, and like the previous example the geometric information contained within each repeat is identical to that of the dual repeat. Also like the previous example this design conforms to the cmm plane symmetry group. These regions of eightfold and sixfold local symmetry surround a central distorted hourglass motif that is produced from underlying hexagons that are unusual in that they have 180° point symmetry rather than reflection symmetry (black circles). Figure 413b is a representation of the design found in a recessed niche in the entry iwan of this complex in Cairo. This is notable in that it is a rare architectural example of a geometric pattern that includes its underlying generative tessellation as part of the overall ornament. It is worth noting that except for the pattern lines that make up the hexagons, the pattern lines are not collinear as they cross the midpoints of the underlying polygons. Changing the angular openings of the pattern lines at the underlying polygonal midpoints is typically avoided, and tends to only be used whenever the pattern elements would not otherwise be well balanced. Such exceptions to the norm are far more common among nonsystematic designs than in patterns created from any of the generative systems.

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Fig. 413

Figure 414 illustrates one of the most elegant nonsystematic designs based upon a rectangular repetitive grid. This is an acute pattern that places 12-pointed stars at the vertices of a rectangular grid, and 10-pointed stars at the vertices of the rectangular dual grid. The 10- and 12-pointed stars are separated by a matrix of 5-pointed stars and mirrored dart motifs. Figure 414a illustrates how these are derived from an underlying tessellation comprised of decagons, dodecagons, pentagons and barrel hexagons. As mentioned previously, the regularity of the pentagons and barrel hexagons from the fivefold system provide ideal conditions for creating patterns in all four families, and the success of this well-balanced pattern is in part the result of the relative regularity of the underlying pentagons—which is to say their closeness to the proportions of the regular pentagon and barrel hexagon of the fivefold system. This design was created by artists during the Seljuk Sultanate of Rum where it was used in the Great Mosque at Aksaray in Turkey (1150-53). It also appears in the anonymous manuscript, On Similar and Complementary Interlocking Figures,Footnote 65 as well as the Topkapi Scroll.Footnote 66 As discussed previously, portions of the anonymous manuscript appear to have been directly influenced by Seljuk geometric ornament, and the rarity of this design suggest the possibility of a link between this manuscript and the example from Aksaray. In addition to the acute pattern in Fig. 414b, the underlying tessellation in Fig. 414a will also produce very acceptable designs in each of the other three pattern families: Fig. 415a shows the median pattern that this underlying tessellation produces; Fig. 415b shows the obtuse pattern; and Fig. 415c shows the two-point pattern. Although not used historically, each of these three patterns (by author) conforms to the aesthetics of this ornamental tradition. Figure 416 demonstrates a construction of the underlying tessellation used for creating the designs in Figs. 414 and 415. Step 1 draws 24 radii within two edge-to-edge dodecagons. Step 2 places a decagon, with 20 radii, in a corner-to-corner arrangement with the two dodecagons. Step 3 mirrors the dodecagons on the vertical radii of the decagon. This establishes the rectangular repeat unit. It is worth noting that this arrangement of dodecagons and decagons can be used for creating patterns, although the non-congruent edges are problematic. Step 4 illustrates the radii matrix that results from the previous steps. Step 5 establishes the edges of the dodecagons and decagons through the standard method using a circle that is tangent to the radii. Step 6 draws the dodecagon and decagon, as well as the two separating pentagons. Step 7 mirrors these elements throughout the rectangle. Step 8 mirrors the trapezoids at the midpoints of the long edge of the rectangle, thus producing the barrel hexagons; and Step 9 shows the completed tessellation. It is worth noting that the Topkapi Scroll’s depiction of the acute design in Fig. 414 includes the underlying tessellation as red lines, and the radii matrix as scribed “dead lines,” and the proportional relationships within these features of the example from the Topkapi Scroll comport with those contained with Fig. 416.

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Fig. 414

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Fig. 415

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Fig. 416

The acute pattern in Fig. 417 is from the entry door of the Abd al-Ghani al-Fakhri mosque in Cairo (1418). This places 10-pointed stars upon the vertices of a rectangular repeat unit, 10-pointed stars at the midpoints of the long edge of the repeat unit, and two 11-pointed stars within the field of the repeat unit. The six trapezoids that have coincident edges with the concave hexagons in Fig. 417a are created by truncating the six pentagons that surround a thin rhombus. As explained earlier, the configuration of six pentagons surrounding a thin rhombus is not suited to the acute pattern family, but, as this example demonstrates, their truncation into six trapezoids allows for very acceptable design features within this family [Fig. 198]. The cluster of three underlying pentagons has also been transformed into three trapezoids surrounding a triangle. The particular proportions of this design cause the trapezoid that is divided by a vertical line of reflective symmetry to be proportionally narrower that the other trapezoids. In order to provide visual balance, the pattern lines of the dart motif associated with this trapezoid extend beyond the midpoints. Similarly, the crossing pattern lines of the dart motifs in the adjacent barrel hexagons are not collinear. Generally, the more eccentric the distortion within the polygons of the underlying tessellation, the greater likelihood of the crossing pattern lines requiring noncollinearity to produce more acceptable results. This is one of the more eccentric geometric designs created by Mamluk artists. Figure 418 demonstrates a construction of the underlying tessellation that creates this Mamluk design. Step 1 places two sets of 20 radii in a vertical orientation. The angle of the two indicated extended radii is 72°. Step 2 places 22 radii near the intersection of the extended radii from Step 1. The choice of 22 radii is determined by the relative closeness of the 72° to the 65.4545…° associated with an 11-fold division of the circle. The precise placement allows the radii connecting the 20- and 22-fold centers to intersect midway between the rotational centers, as indicated by the upper and lower black dots. These radii appear more or less collinear. The precise placement also provides for the intersection of the three red radii at the third black dot. Step 3 mirrors the radii from Step 2 on the indicated vertical and horizontal axes from Step 2. This determines the rectangular repeat unit. The multiple small black dots indicate the intersection points where the radii are not collinear. This completes the radii matrix. Step 4 determines the edges of a decagon using a circle that is tangent with the blue radii. This also determines the size of the hendecagon through simply drawing the circle centered on the end of the blue radius. Step 5 copies the dodecagon to the upper left corner of the repeat, and draws a separating pentagon and barrel hexagon. Step 6 fills in one quadrant of the repeat with the connecting polygonal matrix using the blue lines of the radii matrix. Step 7 mirrors this throughout the repeat. And Step 8 shows the completed tessellation.

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Fig. 417

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Fig. 418

The design in Fig. 419b is from a Mudéjar stucco window grille from Synagogue Transito in Toledo, Spain (1360). This design is a complex arrangement of 6-, 8-, 14-, and 18-pointed stars. Figure 419a demonstrates how this design is comprised of two distinct rectangular repeat patterns: one considerably longer than the other. Either of these can be used independently as a repeat unit, and in this respect, this example can be considered a hybrid design. In this case the shared arrangement of underlying polygons on the short edges of the two repetitive rectangles (18-gons separated by squares) allows them to be used together. Both repetitive rectangles place 18-pointed stars on each corner, and 8-pointed stars within the field of the repeats. The long rectangle also incorporates 14-pointed stars at the center of each repeat unit. Figure 420 shows four stages for a construction of the underlying tessellation for the design from Fig. 419. In the interest of brevity, this sequence is somewhat truncated compared to previous examples. Stage 1 is the radii matrix with 36 radii at each of the corners of both repetitive rectangles, 28 radii at the center of the larger repetitive cell, and 16 radii at appropriate locations throughout the radii matrix. Noncollinearity within the radii matrix is indicated by the black dots. Stage 2 determines the sizes and edge locations of the primary polygons. This standard process makes use of circles placed at intersections within the radii matrix that are tangent to adjacent radii. Stage 3 uses the radii matrix to identify the remaining secondary polygonal elements, including the pentagons, barrel hexagons, and small rectangles. And Stage 4 is the completed tessellation. As mentioned previously, the underlying tessellations of both rectangular repeat units can also be used on their own for pattern generation, although neither are known to have been used on their own within the historical record.

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Fig. 419

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Fig. 420

The acute pattern in Fig. 421 places 12-pointed stars on the corners of the rectangular repeat unit, 8-pointed stars at the midpoints of the long edges of the repeat, and two 9-pointed stars within the field of each repeat unit. This pattern also incorporates octagons within the pattern matrix. This is a Mengujekid design from the Great Mosque of Divrigi in Turkey (1228-29). Figure 422 provides a construction of the underlying tessellation used for creating this design. Step 1 illustrates two sets of radii in a vertical orientation, one with 24 radii, the other with16 radii. Step 2 extends two of the blue radii until they meet. The angle between these extended radii is 52.5° and this is close to the 60° that are 3/18. This provides the opportunity for incorporating a nine-pointed star at this location. Step 3 places 18 radii at the indicated point of intersection from Step 2. These are orientated with the extended line of radius from the 24 radii in Step 2. However, these 18 radii do not align with the relevant lines of radius from the 16 radii. This problem is solved by moving the 18 radii a requisite distance along the black line of radius between the 24 and 18 radii. This distance is determined by copying the black line of radius to the intersection of 24 and 16 red radii, indicated with a black dot. Step 4 trims the red and blue radii that cross the copied black radius from Step 3. Step 5 moves the 18-fold set of radii from the end of the trimmed blue radius to the end of the trimmed blue radius from the 16 radii. These two radii are not collinear but function as such during the process of extracting the underlying tessellation. A black dot indicates the intersection of the two blue radii that are almost collinear. Moving the 18 radii along the line of radius that connects the 24 and 18 radii also connects three of the red radii from these regions of local symmetry. This also provides the necessary conditions within the radii matrix for extracting the underlying tessellation. Step 6 mirrors the radii on the vertical and horizontal dashed red lines indicated in Step 5 to complete the radii matrix. Step 7 determines the edges of the dodecagon, nonagon, and octagon using the standard method of drawing circles that are tangent to the red radii. Step 8 draws the three primary polygons. Step 9 mirrors the primary polygons throughout the repeat. Step 10 fills in the secondary polygonal elements using the radii matrix; and Step 11 illustrates the completed underlying tessellation.

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Fig. 421

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Fig. 422

Figure 423 is a representation of an acute design with 10-pointed stars at the corners of the rectangular repeat unit, 8-pointed stars at the midpoints of the long edge of the repeat, and two 11-pointed stars within the field of each repeat unit. Figure 423a shows the significant disparity between the sizes of the underlying pentagons. This causes the five-pointed stars and adjacent pattern elements in Fig. 423b to vary in size considerably. This, in turn, produces an undulating density throughout the design that is generally less desirable. The discrepancies in the underlying pentagons and barrel hexagons create five- and six-pointed stars that have unsatisfactory characteristics, as are the irregular octagons within the pattern matrix. Another undesirable result of the discrepancies in the sizes of the underlying pentagons is the evident distortion in the proportions of the star rosettes associated with the 11-pointed stars. As a result of these many problems this example lacks elegance and is not a particularly successful geometric design. Were it not for its interesting provenance, these problematic features would have precluded inclusion within this study. This design is from a stone khachkar produced by the Christian monk Momik in Noravank, Armenia, during the thirteenth century. He had presumably received training in this geometric art form from artists working in the neighboring Seljuk Sultanate of Rum, although he does not appear to have mastered this discipline. Figure 424 shows a construction of the underlying tessellation responsible for this Armenian design. Step 1 places 20 radii above 16 radii, and 22 radii at the point of intersection between the two extended blue radii. Step 2 moves the 22 radii to a location that approximates equal conditions in the intersecting radii that connect the 22-fold center with the 20- and 16-fold centers. The two black lines indicate the collinear ideal, while the adjacent red and blue radii indicate the approximate equal conditions between these regions of local symmetry. It is important to note that the fact that these red and blue radii are so far from being collinear indicates the strong likelihood for there being problems in the finished tessellation, as well as any derived patterns. This is the basic flaw that results in all of the many problems with the completed design. Step 3 mirrors these radii to create the radii matrix, and establishes the rectangular repeat. Again, the black lines indicate the collinear ideal. Step 4 places the primary polygons into the radii matrix. Due to the unusually high degree of noncollinearity within the radii matrix, the standard method of circles in tangent with radii does not work well in this case. Rather, the size of the primary polygons has been arrived at through trial and error, with the objective being the least amount of discrepancy between the pentagonal elements. Step 5 fills in the pentagons along the lines of radius. Step 6 adds further pentagons and barrel hexagons. And Step 7 completes the tessellation. The differences between the size and nonconformity of the pentagons (and pseudo pentagons) are readily apparent, and as mentioned produces unavoidable irregularities in any patterns that are created from this underlying tessellation.

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Fig. 423

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Fig. 424

The rectangular acute pattern in Fig. 425 is of a type characterized by linear bands of different primary star forms. In this case the vertical orientation creates an alternating series of 12-, 11-, 10-, 11-, 12-, 11-, 10-, 11-, and 12-pointed stars, etc. The proportion of the rectangular repeat unit for this pattern is unusually long, with 12-pointed stars at the vertices of the repeat unit; 10-pointed stars at the midpoints of the long edge of the repeat; and two 11-pointed stars within the field of the repeat, each located at the approximate centers between the 12- and 10-pointed stars. Figure 425a illustrates the cluster of three pentagons (brown) within the underlying tessellation, as well as an oscillating band of elongated pentagons (pink). These two types of pentagon create very different types of five-pointed stars, and the resulting five-pointed stars in Fig. 425b reflect this discrepancy. As with the pattern in Fig. 423, this discrepancy creates variation within the overall density of the design. However, this example is more cohesive, balanced, and subtle than the previous example, and the variable density is less problematic. This design was created during the Seljuk Sultanate of Rum and is found in the portal of the Erkilet Kiosk near Kayseri, Turkey (1241). Figure 426 illustrates a construction of the underlying tessellation for this pattern. Step 1 shows two sets of 24 radii, one above the other, with extended red radii that has a 60° angle between them. This is close enough to the 65.4545…° angle of the 2/11 division of a circle to provide for an 11-pointed star at this approximate location. Step 2 places an array of 22 radii at the intersection of the two extended red radii from Step 1. This is orientated such that the opening between two of the radii is horizontally aligned, thereby allowing the 22 radii to dialogue equally with the two sets of 24 radii. The fact that the two red radii with the 65.4545…° angle between them are not aligned with the 60° radii from the two sets of 24 radii means that the 22 radii must be moved horizontally to allow these nonaligned radii to intersect at a point that is midway between the 24 and 22 radii centers. To achieve this, two black horizontal lines are drawn at these locations. Step 3 trims the red radii that cross the black lines. Step 4 moves the 22-fold center horizontally so that the two sets of trimmed radii meet at the point indicated by the black dot. These radii appear more or less collinear. Step 5 places two sets of 20 radii such that three blue radii meet at the indicated location (central black dot), thus establishing half of the rectangular repeat. The orientation of these 20 radii is also horizontally aligned. The upper and lower black dots indicate the intersection of what appears to be collinear blue radii. Step 6 determines the edges of the primary polygons through the standard method of circles in tangent with radii. Step 7 draws the dodecagons, hendecagons, and decagons. Step 8 fills in the polygonal matrix using the lines of the radii matrix. And Step 9 shows half of the completed tessellation. This requires mirroring along the indicated axis for the full repeat unit. Like the pattern in Fig. 425, the acute pattern in Fig. 427 is also comprised of 10-, 11-, and 12-pointed stars in a 12, 11, 10, 11, 12, 11, 10, 11, and 12 columnar arrangement. Similarly, this design has an especially long repeat unit with 12-pointed stars at each corner, 10-pointed stars at the midpoints of the long edges of the repeat, and 11-pointed stars between the 10- and 12-pointed stars. These two historical examples share the same repetitive and numeric schema, but the actual patterns are quite distinct from one another. Figure 427a shows the typical six trapezoids surrounding the concave hexagon, and as detailed previously, this arrangement of polygons is particularly well suited to the acute pattern family [Fig. 198]. This is a Zangid design that was reported by Ernst HerzfeldFootnote 67 to have come from a door at the Lower Maqam Ibrahim in the citadel of Aleppo (c. 1230), but is now missing. In noting the highly idiosyncratic conceptual similarities between this Zangid design and the example from the Erkilet Kiosk, and the fact that both are roughly contemporaneous and in relatively close proximity of less than 500 km, it would appear possible that the former of these two geometric designs had a direct influence upon the latter, possibly being created by the same artist or artistic lineage. A construction for the underlying tessellation for the Zangid pattern in Fig. 427 is shown in Fig. 428. As is sometimes the case, a useful starting point is an arrangement of polygons that define the regions of local symmetry. Step 1 shows half of the rectangular repeat unit, with corner-to-corner dodecagons placed at the two corners of the repeat. A hendecagon (11-gon) is placed such that its size and location are determined by two of the corners touching a corner of each dodecagon. Decagons have been placed at the other two corners of the half repeat. These are scaled and located such that their edges are as close as possible to being congruent with the edges of the hendecagon while maintaining their center points upon the same horizontal level as the center points of the dodecagons. Step 2 places radii within each of these polygons. Step 3 shows the finished radii matrix. It is worth noting that this radii matrix is identical to that shown in Fig. 226, and the fact that this can be constructed in more than a single manner is an indication of the flexibility within this design methodology. Step 4 places a new set of dodecagons, hendecagons, and decagons within the set of polygons from Step 1. These are scaled such that a ring of well-proportioned trapezoids surrounds each of the primary polygons. Step 5 shows the completed tessellation.

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Fig. 425

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Fig. 426

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Fig. 427

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Fig. 428

3.2.6 Hexagonal Designs with Multiple Regions of Local Symmetry

There are relatively few historical nonsystematic patterns with multiple regions of local symmetry that repeat upon a hexagonal grid. Yet several of the examples that adhere to this repetitive schema are among the most geometrically interesting designs within this ornamental tradition. In particular is small set of patterns that employ the principle of adjacent numbers wherein the two regions of local symmetry are one numeric step above and one numeric step below an n-fold symmetry that works conveniently on its own to fill the plane. In this way, just as eight-pointed stars are predisposed to work on their own, so also should patterns with seven- and nine-pointed stars. Similarly, patterns with just 10-pointed stars suggest the possibility for those with 9- and 11-pointed stars, and those comprised of 12-pointed stars anticipate patterns with 11- and 13-pointed stars. Each of the following historical examples that employ the principle of adjacent numbers repeats upon an elongated hexagonal grid that places one of the two types of primary star at each of the vertices of the repetitive grid. The dual of these grids are also comprised of elongated hexagons, the vertices of which have the other primary star form. The proportion of each grid is determined by the n-fold symmetry of the primary stars, and the dual grids are perpendicularly orientated from one another.

Figure 429d represents a Seljuk border that surrounds the mihrab of the Friday Mosque at Barsian, near Isfahan (1105). This is a median pattern that juxtaposes seven- and nine-pointed stars that repeat upon either of two perpendicular elongated hexagonal grids indicated in Fig. 429e. Figures 429a and b demonstrate the principle of adjacent numbers wherein a polygon that tessellates easily on its own—in this case the octagon—indicates the ability of polygons that are one numeric step above and below to also tessellate successfully—in this case the nonagon and heptagon. Note: While the four octagons that cluster around the central square have congruent edges, the edges of the two regular nonagons and two regular heptagons that cluster around the central square, while congruent with the edges of the square, do not quite align with one another (although they are close). Figure 429c shows the linear arrangement of these two underlying polygons that produces the border design at Barsian. Figure 429d demonstrates how extending the width of the border past the linear line of symmetry that divides the underlying heptagons in half disguises the presence of the seven-pointed stars. Rather than being truncated along the line of symmetry at the edge of the border, the pattern continues slightly past the linear line of symmetry to create a wider border, and designated pattern lines at the edges of the border are woven back into the design. In this example from Barsian, some of the unresolved pattern lines along the edge are extended beyond the border to become part of the adjacent kufi calligraphy (not shown). Figure 429e illustrates the extension of the linear underlying tessellation from Fig. 429c to a full tessellation that covers the plane. This illustration also shows the two perpendicularly orientated hexagonal dual grids that provide the repetitive structure. The underlying nonagons in this arrangement are conjoined, and unlike the superb historical example from Barsian, the pattern lines in this region are not full nine-pointed stars. The design in Fig. 429f (by author) creates an interweaving version of the design in Fig. 429e. While acceptable, this would be more pleasing were it to have full nine-pointed stars. This design (sans interweave) conforms to the pmm plane symmetry group, and shares visual characteristics with the classic star and cross pattern created from the tessellation of octagons and squares. Figure 430a demonstrates how the arrangement of heptagons and overlapping nonagons from Fig. 429e can be used to produce a tessellation wherein the primary polygons are surrounded by a polygonal matrix of pentagon and barrel hexagons. This further development has the advantage of full nonagons that do not overlap, thereby allowing for non-interrupted nine-pointed stars within any resulting patterns. Figure 430a shows the sequence of producing this secondary new underlying tessellation. This sequence begins with the creation of a radii matrix, from which the polygonal matrix is easily derived. Figure 430b shows an acute pattern (by author) created from this tessellation. The four clustered pentagons distributed throughout the underlying tessellation provides for the octagons within the pattern matrix. These octagons and the pattern lines surrounding the octagons have shared qualities with some acute designs created from the fourfold system B [Fig. 177a]. The repetitive structure of this design is identical to that of Fig. 429f, with two varieties of perpendicularly oriented hexagonal grids.

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Fig. 429

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Fig. 430

The acute pattern in Fig. 431 has the same dual hexagonal repetitive structure as the examples from Figs. 429 and 430. However, in this case the included angles of the hexagonal repeat units are derived from 9- and 11-fold local symmetries. The horizontally orientated hexagonal grid has 11-pointed stars at the vertices, and the vertically orientated dual-hexagonal grid has 9-pointed stars at the vertices. This remarkable pattern is from the Topkapi Scroll,Footnote 68 but is currently unknown to the architectural record. As per the principle of adjacent numbers, the occurrence of 9- and 11-pointed stars is presupposed by the convenience of 10-pointed stars for covering the plane. What is more, the layout of the 5-pointed stars and facing darts that separate the 9- and 11-pointed stars bears remarkable similarity to that of the classic fivefold acute design [Fig. 226]. Figure 432 illustrates a construction of the underlying tessellation responsible for this design from the Topkapi scroll. Figure 432a shows how nonagons and hendecagons can be arranged around a concave hexagon in a very similar manner as decagons around a concave hexagon. Unlike the decagons, the edges of the nonagons and hendecagons are not quite contiguous. The ability of these polygons to work together in this fashion is an expression of the principle of adjacent numbers wherein the 9- and 11-sided polygons are able to cover the plane in a similar manner as the 10-sided polygon is. Figure 432b illustrates this two-dimensional coverage, along with the radii matrix that this arrangement facilitates. Figure 432c shows just the radii matrix, along with black dots that indicate locations were the radii are not quite collinear. Figure 432d illustrates the radii matrix along with a polygonal tessellation that is created from the radii matrix. This is produced using the standard formula of circles in tangent with the radii that has been detailed previously. Figure 432e illustrates this new tessellation with hendecagons at the vertices of the horizontally orientated hexagonal grid, and nonagons at the vertices of the vertically orientated dual hexagonal grid. Figure 432f shows this underlying tessellation along with the acute pattern that is produced from this tessellation. The shaded region is the minimal rectangular repetitive cell, and represents the portion of this design illustrated in the Topkapi Scroll. It is worth mentioning that along with the pattern itself, the illustration in the Topkapi Scroll accompanies the design with both the radii matrix and the underlying tessellation. Figure 433 shows an alternative construction sequence for the underlying tessellation for the acute pattern from the Topkapi Scroll in Fig. 431. Step 1 is an array of 22 radii. Step 2 mirrors the 22 radii. Step 3 mirrors the 22 radii again. Step 4 shows half of the hexagonal repeat unit along with the extended radii from each corner. The indicated angle is 40.9091…°. This is close to the 40° angle associated with a ninefold division of the circle, and indicates that a nine-pointed star should be able to work at this location. Step 5 places 18 radii at the intersection of the four blue radii from Step 4. These radii can be seen to be out of alignment with the four blue radii that originate at each corner and meet at the center of the trapezoid. The black horizontal line indicates the small amount of horizontal movement of the 18-fold center to make these blue radii intersect. Step 6 shows the results of this move. The radii that connect the 22-fold and 18-fold centers are not quite collinear, but function as though they are. The intersections are indicated in black dots. Step 7 determines the edges of the nonagon and hendecagon through the placement of a circle that is tangent with the blue radii. Step 8 draws these primary polygons, as well as the two separating pentagons. Step 9 mirrors the hendecagons to each corner of the half repeat, and fills in the half repeat with further pentagons and barrel hexagons. And Step 10 is the completed tessellation. It is worth mentioning that this underlying tessellation will make very acceptable designs in all three of the other pattern families.

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Fig. 431

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Fig. 432

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Fig. 433

The pattern in Fig. 434 is from the Mu’mine Khatun mausoleum in Nakhichevan, Azerbaijan (1186). This remarkable Ildegizid pattern is comprised of 11- and 13-pointed stars [Photograph 35]. Figure 434a illustrates how the underlying hendecagons are located at the vertices of a horizontally orientated hexagonal grid, and the tridecagons are placed on the vertices of a vertically orientated dual hexagonal grid. Both of these hexagonal grids have translation symmetry and can be regarded as the repeat unit. The turquoise hendecagons and tridecagons that surround the stars in Fig. 434b are an arbitrary addition on the part of the artist. The overall balance of this design is aesthetically pleasing, and this non-challenging quality belies the considerable geometric complexity that underlies this construction. The principle of adjacent numbers is also a feature of this Ildegizid pattern. Figure 435a demonstrates how the square and triangular arrangements of the dodecagon have analogous arrangements with the hendecagon and tridecagon. Of course the proportions of the square and regular triangles change according to the angles associated with the 11- and 13-fold division of the circle. Figure 435b places hendecagons and tridecagons into an arrangement wherein their edges are very close to being contiguous. This also places a radii matrix of 22- and 26-fold rotational centers within these figures. Figure 435c shows the radii matrix alone, with added black dots that indicate the intersection of radii that appear collinear. Figure 435d shows the new tessellation with connecting pentagons, along with the generative radii matrix. Figure 435e shows just the new tessellation along with the two perpendicular dual hexagonal grids. And Fig. 435f shows the underlying tessellation with the acute pattern from Nakhichevan. Figure 436 illustrates the analogous relationship between the underlying tessellation for the pattern in Fig. 434 and a hybrid tessellation of dodecagons placed upon the vertices of a grid comprised of squares and equilateral triangles. The tessellation in Fig. 436a has vertical and horizontal lines of reflected symmetry (dashed lines), and the full rectangular panel has translation symmetry. As a repeat, this rectangle has twice the area of either of the hexagonal repeat units. The point of intersection of the red diagonal lines, where four pentagons meet, has 180° rotation symmetry. The square panel in Fig. 436b also has two lines of reflected symmetry, and also has translation symmetry. Rather than 180° rotation symmetry, the analogous location were four pentagons meet, and the red diagonal lines intersect has 90° rotation symmetry. The visual similarity between these two tessellations is especially apparent in the respective arrangements of the matrix of pentagons that separate the primary polygons. The underlying tessellation for the pattern in Fig. 434 will also make very acceptable designs in the other three pattern families. Each of the three designs (by author) in Fig. 437 is comprised of 11- and 13-pointed stars, and repeats on either of the two hexagonal grids. Figure 437a shows a median pattern that has been modified in the common technique often found in Mamluk designs [Fig. 223]. Figure 437b shows an obtuse pattern. And Fig. 437c shows a two-point pattern with a standard variation to the primary stars [Fig. 225b]. Figure 438 demonstrates a construction of the underlying tessellation for the pattern in Fig. 434. Step 1 mirrors 22 radii so they have a vertical orientation. Step 2 mirrors these 22-fold centers as shown, thereby identifying the trapezoid (blue) that is half the hexagonal repeat unit. The indicated angle is 81.8181…°. This is close to the 83.0769…° associated with 6/26 of a 26-fold division of a circle. Step 3 places 26 radii at the intersection of the extended blue radii from Step 2. Four of the 26-fold blue radii do not quite align with the associated 22-fold radii. To correct this non-alignment two black horizontal lines are placed for moving the 26-fold center and for trimming the blue radii. The slight movement of the 26-fold center must be horizontal so that the final resting place remains central within the repeat. Step 4 shows the radii matrix after moving the 26-fold center so that the trimmed blue radii of the 26-fold center meets the trimmed blue radii of the 22-fold centers. The black dots indicate the intersections of the trimmed blue radii that connect the 22- and 26-fold centers. These appear collinear, but actually have slight angles off of 180°. Step 5 determines the edges of the hendecagons and tridecagon using a circle that is tangent to the blue radii. Step 6 draws the two primary polygons. Step 7 mirrors the hendecagons into the other three corner locations. Step 8 uses the lines of the radii matrix to fill the half repeat with the matrix of connecting pentagons. Step 9 mirrors the half repeat; and Step 10 shows the completed tessellation within its hexagonal repeat unit.

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Fig. 434

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Fig. 435

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Fig. 436

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Fig. 437

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Fig. 438

Hexagonal nonsystematic patterns with more than one region of local symmetry do not have to conform to the repetitive schema associated with the principle of adjacent numbers—as per the previous three examples. Figure 439 shows an acute pattern that repeats upon a hexagonal grid, with 9-pointed stars placed at the vertices of the repetitive grid, 12-pointed stars at the midpoints of the aligned parallel edges, 8-pointed stars at the centers of each repeat unit, and four 10-pointed stars within the field of each repeat unit. There are also octagons within the pattern matrix. As seen in Fig. 439a, there is substantial irregularity in the sizes and shapes of the polygons that make up the connecting matrix, and this transfers to the significant disparity and inconsistency within the pattern matrix of Fig. 439b. This design originates from the Seljuk Sultanate of Rum and is found in the courtyard portal of the Karatay Han near Kayseri, Turkey (1235-41).

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Fig. 439

3.2.7 Radial Designs with Multiple Regions of Local Symmetry

Nonsystematic geometric patterns with radial symmetry are rare within the ornamental arts of Muslim cultures. The obtuse pattern in Fig. 440 dates from the Safavid period, and is from one of the star shaped soffits within the muqarnas of the southeast iwan (fifteenth to sixteenth century) at the Friday Mosque in Isfahan. The soffit is a seven-pointed star, and the obtuse design contained within this soffit has sevenfold rotational symmetry. A 14-pointed star is located at the center of the 7-pointed star, and 11-pointed stars are placed at the inside angles of the 7-pointed star. The furthest angle of the seven-pointed star has a partial nine-pointed star. Figure 440 demonstrates how this design can be created from either of two underlying tessellations. In a similar fashion as the two tessellations in Fig. 200, these underlying tessellations have a dual relationship.

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Fig. 440

Figure 441 shows a design also from a soffit in the muqarnas semidome of the southeast iwan in the Friday Mosque at Isfahan [Photograph 90]. This is a radial design with tenfold rotation symmetry set within a ten-pointed star. This median pattern places a ten-pointed star at the radial center, and 10 ten-pointed stars at the outer obtuse angles of the ten-pointed star panel. Ten 7-pointed stars surround the central 10-pointed star, and partial 7-pointed stars separate the 10-pointed partial stars at the periphery of the panel. The heptagons that form the ring in Fig. 441a are not regular, and the seven-pointed stars do not have sevenfold rotation symmetry. In Fig. 441a, the decagons located at the 108° included angles of the points of the ten-pointed star are separated by very irregular heptagons. The visual characteristics of this median pattern have distinct parallels with median patterns created from the fourfold system A, for example, Figs. 154 and 159.

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Fig. 441

3.3 Dual-Level Designs

The tradition of dual-level geometric patterns with self-similar characteristics is one of the most striking developments in the long and multifaceted history of Islamic geometric ornament. The secondary application of a smaller scale geometric design into an otherwise independent larger scale primary pattern was highly innovative, and the development of dual-level patterns was the last great geometric and aesthetic advance of this artistic tradition. The multi-level quality of this variety of historical Islamic geometric design has characteristics that invite comparison to several areas of contemporary mathematics and crystallography. What is more, and as per the examples at the end of this section, the methodological practices that were employed by the originators of these dual-level designs can be used to produce designs that accurately conform to some of these recent mathematical and crystallographic discoveries. Not all readers will be familiar with the technical terminology employed within this field, and definitions have been provided in the glossary section of this book to help facilitate a more precise understanding of such concepts as self-similarly, aperiodicity, quasiperiodicity, quasicrystallinity, subdivision rules, and Penrose matching rules.

The mature style of dual-level design developed in the Maghreb during the fourteenth century, and approximately a century later in Persia, Khurasan, and Transoxiana. Precursors to the fully mature style are found in several locations in the eastern regions, including the exterior façade of the Gunbad-i Qabud in Maragha, Iran (1196-97) [Fig. 67] [Photograph 24]; the minaret of the Yakutiye madrasa in Erzurum, Turkey (1310); and the mausoleum of Uljaytu in Sultaniya (1307-13) [Photograph 96]. In the Maghreb, the incorporation of dual-level designs was implemented as a fully mature tradition from its onset during the fourteenth century, indicating that it may have been innovated by a single person or atelier rather than part of an ongoing developmental process. The mature style of dual-level design in both the eastern and western regions is characterized by the almost universal use of three-, four-, and fivefold polygonal systems. While appearing to have considerable complexity, the systematic basis behind their design methodology is surprisingly accessible, and their increased complexity stems from the geometric elaboration of a common systematic theme rather than increased symmetrical elaboration.

In creating dual-level designs, the use of polygonal systems such as the system of regular polygons, the fourfold system A, and the fivefold system provides proportional continuity between a design’s primary and secondary levels. In practice, this involves the application of scaled-down secondary underlying polygonal modules, along with their associated pattern lines, to the primary pattern lines and primary background elements. Theoretically, scaled-down modules with their associated pattern lines can be applied infinitely,Footnote 69 but all of the examples from the historical record have only two levels. Yet, even with just two levels, these designs occasionally fulfill the mathematical criteria for self-similarity.

Figure 442 illustrates the four historical varieties of dual-level design. For comparison, each of these examples (by author) employs the same classic fivefold obtuse pattern as the primary pattern. Type A dual-level designs emphasize the primary pattern with a single plain line upon which the secondary pattern is constructed. Both the primary and secondary patterns are uninterrupted, and fill the repetitive cell to the full extent. As with this example, the primary stars of the secondary pattern are typically located upon the intersections of the primary pattern. Type B dual-level designs are characterized by widening the lines of the primary pattern and infilling the widened lines with a secondary pattern. As in this example, this variety of dual-level design typically places the primary stars of the secondary pattern at the vertices of the widened primary pattern. The background regions are either left plain (as in this case) or provided with a floral or occasionally calligraphic design. As with type B designs, type C dual-level designs also widen the primary pattern and apply the secondary pattern into the widened lines. However, this variety of dual-level design extends the secondary pattern to fully cover the repetitive cell. The bounding lines of the primary pattern are maintained, but the differentiation between the primary and secondary patterns is emphasized through the use of color. Type D dual-level designs use color as the only method of differentiating the primary and secondary patterns. In this variety of dual-level design the primary pattern is exposed through picking out the appropriate background regions of the secondary pattern. Without the color differentiation, type D dual-level designs would appear as particularly complex single-level geometric designs. Types A, B, and C are native to Persia, Khurasan, and Transoxiana, with types A and B being most common and type C being comparatively rare. Type D dual-level designs are exclusively found in Morocco and al-Andalus.

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Fig. 442

Figure 443 demonstrates the type A application of the secondary pattern to the primary pattern through detailing the example illustrated in Fig. 442. This example employs the fivefold system, but the same basic methodology also works with each of the other historical design systems. As shown, the primary pattern in this example is comprised of two intervals that have the phi [φ] proportions of the golden ratio. Figure 443a determines the scale of the secondary pattern by centering two edge-to-edge decagons at each end of the longer interval. As shown in Fig. 447, other arrangements of underlying polygons can be used, but the key to this design methodology is placing the scaled-down polygonal modules so that they fit within the intervals of the primary pattern. Figure 443b applies the scaled-down decagons to other intersections throughout the primary pattern. Figure 443c fills in the remaining background with additional modules from the same system to complete the secondary underlying tessellation. And Fig. 443d applies pattern lines associated with the median family to complete the design (as per Fig. 442: type A). Although less common, some historical type A dual-level designs place the primary polygons of the secondary pattern at the intersections of the primary pattern as well as the vertices of the primary underlying tessellation. Figure 444a places two edge-to-edge decagons such that one is placed upon the primary pattern (black) and the other is placed on the primary underlying tessellation (red). These two locations are the closest points between these two varieties of vertex. Figure 444b places these scaled-down decagons at all the intersections of the primary pattern and all the vertices of the primary underlying tessellation. Figure 444c populates the edges of the primary pattern and the edges of the primary underlying tessellation with further polygonal modules from the same system. Figure 444d fills in the remaining background to complete the secondary underlying tessellation. Figure 444e illustrates the application of the secondary obtuse pattern to the secondary underlying tessellation; and Fig. 444f shows just the pattern lines that make up the dual-level design (by author) created from this tessellation.

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Fig. 443

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Fig. 444

A primary concern in creating type B and type C dual-level designs is determining the proportions of the widened primary design. By widening the lines so that the proportions adhere to the proportions of the generative system—in this case the fivefold system—the widened lines will allow for the scaled-down modules from the same system to fit precisely into the widened region. Figure 445a demonstrates how an appropriate proportion for the widened lines can be determined easily by employing modules from the fivefold system: in this case the wide rhombus and the trapezoid placed in a pentagonal configuration. The use of these modules insures that the intervals within the widened lines have the phi proportions of the golden section that are required for adding the scaled-down polygonal modules when working with the fivefold system. Figure 445b places decagons at the intersections of the widened lines, and the scale of these decagons is determined by their edge-to-edge placement throughout this network. Figure 445c shows the type B variation with the trimmed away secondary modules so that only the widened lines are populated with both the underlying tessellation and the applied pattern lines of the median family. Figure 445d through f are type C dual-level designs. Figure 445d shows the same widened primary lines and the same applied secondary modules as Fig. 445c. However, the secondary underlying tessellation is extended throughout the repetitive cell, thereby infilling the background regions with additional polygonal modules. Figure 445e applies the median pattern lines to this tessellation; and Fig. 445f illustrates the completed type C dual-level design. It is interesting to note the pattern line conditions within the decagons that are located within the widened primary design, but not centered on a vertex of the primary design. These are near to the inside corner of the primary ten-pointed stars. The fact that these secondary decagons are not centered on the primary vertices, and that the point at which their edges cross the lines of the primary design are not the midpoints of the secondary decagon (as is the case with the decagons that are placed on the vertices of the primary widened design), might lead one to expect that the applied secondary pattern lines inside these decagons would be arbitrary, and would not necessarily work well with the primary widened pattern lines. However, a careful look at the applied pattern lines to these secondary decagons in Fig. 445f reveals that the interior points of the secondary ten-pointed stars rest precisely upon the lines of the primary design. This type of concordance in intersection points between the applied pattern lines of multiple levels of design is a remarkable, albeit standard feature of the recursive use of these design systems, and is the result of the proportional continuum that is inherent throughout the multiple levels of scaled-down polygonal modules and their resulting pattern lines.

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Fig. 445

The method for creating type D dual-level design from polygonal systems is essentially the same as that of the type A designs, with one important proviso: the rotational orientation of the primary polygons for the secondary pattern—in this case scaled-down decagons—must allow for the creation of secondary pattern elements that express the primary pattern through color differentiation. Figure 446a illustrates how this is achieved by placing the decagons such that two sets of the applied secondary pattern lines that make up the ten-pointed stars run parallel with the directions of the primary pattern. The requirement of secondary pattern lines that are parallel with the primary pattern lines also makes the choice of pattern family critical to this process. Rotational orientation and suitable pattern family are required regardless of the polygonal system that is being used. Once the initial application of suitably orientated secondary polygonal modules has been applied to the vertices of the primary pattern, the infill of the remaining areas is completed. Figure 446b shows a completed secondary tessellation with overlapping decagons. Figure 446c shows how these overlapping decagons can be transformed to allow for the placement of multiple 1/10 triangular segments of the decagon and large rhombi. These configurations produce good results with the acute family [Fig. 196]. Figure 446d shows the acute secondary pattern, along with the primary pattern that is differentiated through the use of color within the secondary background elements.

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Fig. 446

Figure 447 illustrates a selection of alternative secondary polygonal arrangements for populating the same pentagonal region that represents the primary pattern lines. This simple process involves first determining the edge configuration, followed by filling in the remaining interior of the primary pentagon with further polygonal modules. Invariably, the scale factor between levels is governed by the secondary polygonal edge configuration, and the precise scale will always be an expression of the inherent proportions of the generative system. The examples in this figure are produced from the fivefold system, and the consequent scale factors are expressions of phi. The column on the left side of this illustration indicates two edges of the primary underlying generative decagon in green, and the primary pentagonal pattern lines of the obtuse family in black. The provided scaling ratio is between the length of the secondary polygonal edges (red) and the primary decagonal edges (green). The edges of the primary polygonal modules in the column on the right are the green pentagons, and the primary pattern lines are the black pentagons that contact the midpoints of the green pentagons. Once again, the provided scaling ratio is between the length of the secondary polygonal edges (red) and the primary decagonal edges (green). In all eight of these examples the scaling ration is indicated as an expression of phi.

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Fig. 447

3.3.1 Historical Examples of Type A Dual-Level Designs

Figure 448 is a dual-level design from a Mughal pierced marble jali window grille at the I’timad al-Daula in Agra, India (1622-28). Figure 448a illustrates the two generative tessellations of regular hexagons that produce both levels of design in Fig. 448b. Taken on their own, the design of both the primary and secondary levels is the classic six-pointed star median design created from the system of regular polygons [Fig. 95b]. The proportional scale between the primary and secondary hexagonal grids corresponds with 2 × √3, or 1:3.4610…. Figure 448b applies six-pointed stars to the midpoints of each primary and secondary underlying hexagon. Figure 448c illustrates the dual-level pattern without its underlying generative tessellation. Peter Cromwell has observed that this example is the only known historical Islamic dual-level geometric design with scale invariance wherein the primary and secondary patterns are identical, but for their scale and 90° rotational orientations.Footnote 70 Geometric structures with scale invariance such as this Mughal example from India have a high degree of self-similar symmetry. The scale invariance of this example applies equally to both the underlying generative tessellation and the resulting dual-level design. This exactitude is distinct from the looser forms of self-similarity of other historical dual-level designs wherein the primary and secondary patterns are both created from polygonal modules contained within the same modular system, and where both the primary and secondary patterns are of the same pattern family, but where the two respective patterns are not otherwise identical. Still less precise are those historical dual-level patterns were the primary and secondary patterns are created from the same generative system, but the primary design and secondary design are produced from different pattern families.

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Fig. 448

Of the many dual-level designs illustrated in the Topkapi scroll, number 31Footnote 71 has scale invariance within a limited region of the overall repeat unit. Both the primary and secondary patterns of the type A dual-level design in Fig. 449 are of the median family, and created from the fivefold system. The overall rectangular repeat of the primary design is recursively iterated at the center of the secondary pattern. As indicated in Fig. 450d, this rectangular arrangement of 4 ten-pointed stars is placed upon four of the vertices of the central hexagon in the primary design. The only difference between the scaled-down use of the primary design is in the treatment of the ten-pointed stars: the scaled-down ten-pointed stars in the example from the Topkapi Scroll have ten kites in rotation, while the primary stars are without kites. As indicated, this scaled-down repetitive rectangle is also used to the right and left of the central location. Aside from these iterative regions, the secondary pattern is not totally the same as the primary pattern, and the self-similarity conforms to the use of median patterns created from the same methodological system at two scales. Note: The example from the Topkapi Scroll is black lines only, and the color in this illustration has been added for visual clarity. Figure 450 demonstrates the method of creating the secondary pattern by applying scaled-down polygonal modules from the same fivefold system to key locations of the primary pattern. These scaled-down polygonal modules are present in the scribed “dead lines” of the Topkapi Scroll. Figure 450a shows a rectangular repeat unit with the underlying tessellation of decagons, pentagons and wide rhombi and the primary median pattern created from this tessellation. Figure 450b places decagons with applied pattern lines of the median family upon intersections of the primary pattern. These secondary decagons are scaled such that they fit in an edge-to-edge arrangement at the 72° and 108° angles of the primary pattern. The leftover regions are filled in Fig. 450c. The inherent phi proportions of the fivefold system insure that the scaled-down polygonal modules will seamlessly tessellate the remaining areas between the secondary decagons. The scale factor for this dual-level design is 1:5.2360…. and the relation to phi is 2 + [(1 + √5 ÷ 2) × 2], or more simply as 2 + [φ × 2]. As mentioned, the secondary pattern and secondary tessellation in each of the three shaded rectangles at the center of Fig. 450d are identical to the rectangular repeat, underlying tessellation and primary pattern in Fig. 450a (except for the addition of the kite elements within the ten-pointed stars).

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Fig. 449

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Fig. 450

Figure 451 represents a Qara Qoyunlu cut-tile mosaic type A dual-level design in one of the blind arches at the Imamzada Darb-i Imam in Isfahan (1453) [Photograph 97]. This design is repeated within an arched spandrel at the Imamzada Darb-i Imam, the only difference being the colorization, and slight variations within the secondary infill of the primary ten-pointed stars. The primary pattern in this example is the classic fivefold obtuse design [Fig. 229a], and the secondary pattern is also from the obtuse family. While both the primary and secondary patterns are created from the fivefold system, the primary pattern is not replicated within the secondary pattern. For this reason, there is no scale invariance present in this example. Therefore, the more loosely defined self-similarity in this dual-level design is a product of its using polygonal modules from the same fivefold system at both scales, as well as the same pattern family at both scales. The secondary pattern of this historical example has some anomalous properties that warrant examination. Specifically, the secondary pattern contained with the pentagons of the primary pattern have neither reflected nor rotational symmetry. Rather, the polygonal infill of the pentagonal regions eschews these more conventional forms of symmetry in favor of a pattern that, at first glance, appears to have rotation symmetry, but breaks this around the periphery of the pentagonal infill. As pointed out by Peter Cromwell,Footnote 72 the secondary pattern within each of the primary pentagons has the exact same anomalous pattern, but with different rotational orientations within each respective pentagon. It has been suggested that the non-rotational symmetry of the secondary design contained within the primary pentagons was a mistake on the part of the artist, perhaps introduced while the original panel was being repaired.Footnote 73 However, the occurrence of this unusual feature within each of the primary pentagonal elements would appear to indicate a willful intent. This is further confirmed by the fact that the primary pentagonal regions of the dual-level design from the nearby arch spandrel at the Imamzada Darb-i Imam also employs the same unusual arrangement of secondary design elements within the primary pentagonal regions. To add to this refutation, both of these dual-level examples at the Imamzada Darb-i Imam are in excellent condition and show no signs of having been repaired. The anomalous secondary pattern treatment within the pentagonal regions interrupts the rotational symmetry that would otherwise be a standard feature of these two examples. Additionally, this anomalous treatment of the secondary pattern within the primary pentagonal regions interrupts the reflection symmetry that would otherwise be present, thereby changing the plane symmetry group for both these dual-level designs to p1 rather than pmm. In 2007 Peter Lu and Paul Steinhardt claimed to have discovered “nearly perfect” quasicrystalline tilings in the ornamental mosaics at the Imamzada Darb-i Imam.Footnote 74 Despite the significant media attention such claims generated, there are serious problems with their arguments. Their claim that “the Darb-i Imam tessellation is not embedded in a periodic framework and can, in principle, be extended into an infinite quasiperiodic pattern” is contradicted by the fact that this dual-level design has very obvious translation symmetry. As indicated by the red diagonal lines in Fig. 451, this dual-level design repeats upon a rhombic grid, and it is immaterial that the secondary pattern is comparatively complex. Seeking to demonstrate quasicrystalline aperiodicity by only examining an isolated region of the secondary pattern, and comparing this to Penrose tilings with subdivision rules, ignores the fact that the isolated region is unquestionably part of a larger periodic structure.Footnote 75 Their article created considerable debate over the merits of their claim of quasicrystallinity within the design at the Darb-i Imam. A persuasive counter argument has been made by Peter Cromwell who details the flaws in their claim to have identified Penrose subdivision rules within the example from the Darb-i Imam.Footnote 76 Lu and Steinhardt are correct in identifying the self-similar characteristics between the primary and secondary patterns, but they were not the first to discover recursive self-similarity within this tradition,Footnote 77 or even at the Imamzada Darb-i Imam.Footnote 78 Nor were they the first, as claimed, to recognize how the secondary pattern is the product of a set of underlying polygonal modules, or the correlation between the use of these modules in both this design from Isfahan and examples from the Topkapi Scroll.Footnote 79 Similarly, these authors are correct in identifying the potential of the fivefold system (although they do not use this prior terminology) for making Islamic geometric patterns that are true quasiperiodic structures devoid of translation symmetry. Here again, they were not the first to identify this remarkable capability of the fivefold system.Footnote 80

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Fig. 451

Figure 452a shows the classic fivefold obtuse design along with its underlying generative tessellation. Figure 452b applies two secondary underlying tessellations to the primary pattern in Fig. 452a. As is common within the fivefold system [Fig. 200], these have a dual relationship and either can be used to create the secondary obtuse pattern from this historical example. Figure 452c demonstrates the method of determining the size of the scaled-down secondary polygonal modules: edge-to-edge decagons are placed such that their centers rest upon the closest interval in the primary design. Decagons are then placed at each intersection of the primary design, and these are connected with concave hexagons along the relevant primary pattern lines. Figure 452d completes the underlying tessellation by applying further decagons, concave hexagons and long hexagons into the unfilled background regions from Fig. 452c. The initial layout of the alternative underlying secondary tessellation that produces the secondary pattern is demonstrated in Fig. 452e. This places decagons that are separated by two contiguous pentagons at the shortest interval of the primary design. The continued population of the primary pattern lines with scaled-down polygonal modules simply copies the secondary decagons and pentagons to each vertex of the primary pattern, and the gaps are conveniently filled with barrel hexagons from the fivefold system. Figure 452f fills the undeveloped areas of Fig. 452e with additional pentagons, barrel hexagons and thin rhombi of the fivefold system. The scale factor for this dual-level design is 1:8.4721… which can be expressed as 2 + [φ × 4], or as 4 + [√5 × 2].

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Fig. 452

Figure 453 shows a Safavid cut-tile mosaic type A dual-level design from the Madar-i Shah in Isfahan (1706-14) that is also created from the fivefold system. The primary pattern is the classic acute design found throughout the Islamic world, and the secondary level is an obtuse pattern. This is a looser form of self-similarity that applies only to the design methodology wherein the same underlying polygonal modules from the fivefold system are employed recursively at differing scales, but does not apply to the visual result wherein two distinct pattern families are used at different scales. Needless to say, the use of two different pattern families at both levels precludes the possibility of scale invariance within the design. Figure 454a shows the classic fivefold acute design along with its underlying generative tessellation. Figure 454b shows two secondary grids with dual characteristics, either of which can be used to create the secondary obtuse pattern from this dual-level design. Figure 454c illustrates the applied secondary decagons, concave hexagons, and long hexagonal modules to the intersections of the primary design, as well as to the vertices of the underlying tessellation for the primary design. The scale of the secondary modules is determined by making the centers of two edge-to-edge decagons equal the shortest distance between intersections in the primary design. Figure 454d shows the primary and secondary designs along with the primary and secondary underlying tessellation. Figure 454e shows the alternative underlying tessellation for producing the secondary obtuse pattern. This is comprised of decagons, pentagons, barrel hexagons, and thin rhombi. The decagons are likewise placed at the intersections of the primary design as well as the vertices of the primary underlying tessellation. The scale of the secondary modules is determined by a configuration of two decagons separated by two pentagons placed at the shortest interval of the primary design. Figure 454f illustrates the completed design along with the primary and secondary underlying tessellations. The scale factor between the two levels is 1:13.7082… which can be expressed as 4 + [φ × 6], or as 7 + [√5 × 3].

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Fig. 453

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Fig. 454

3.3.2 Historical Examples of Type B Dual-Level Designs

Figure 455a illustrates a Janid type B dual-level design from the Nadir Divan Beg madrasa in Bukhara (1622-23) [Photograph 100]. As stated, this variety of dual-level design widens the primary pattern lines and fills this widened region with the secondary design. Figure 455b illustrates the simplicity of the primary design: a linear band of hexagons and triangles, with the widened lines determined by the governing isometric grid. The secondary design is from the obtuse family, and is comprised of six- and nine-pointed stars, with the six-pointed stars placed at the vertices of the governing isometric grid. As shown, the design and width of the widened lines of the primary pattern are easily derived from the isometric grid, but can also be identified as the 3.6.3.6 semi-regular tessellation [Fig. 89]. The secondary pattern is nonsystematic, and works within this structure by virtue of the triangular repetitive cells. Figure 456a shows the secondary underlying tessellation of nonagons surrounded by pentagons, with six-pointed star interstice regions at each vertex of the repetitive isometric grid. Figure 456b shows the secondary obtuse pattern created from the underlying tessellation. The pattern lines within each underlying six-pointed star interstice region are an arbitrary treatment that is not determined by the underlying tessellation. The secondary design on its own is essentially identical to several single-level examples from the same region, including an earlier Shaybanid pattern at the Kukeltash madrasa in Bukhara (1568-69) that may have served as an inspiration for this dual design [Fig. 313a].

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Fig. 455

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Fig. 456

Figure 457a represents one of the two very similar Timurid type B dual-level designs from the Friday Mosque at Varzaneh near Isfahan (1442-44). As per the previous example, Figure 457b demonstrates how the primary widened lines can be easily derived from the isometric grid. The primary design is one of the most basic threefold patterns, and can be easily created from the isometric grid (as shown), or from an underlying tessellation of just hexagons [Fig. 95b]. Figure 457b illustrates the underlying tessellation that creates the secondary pattern. This is comprised of dodecagons and triangles with the dodecagons placed at the vertices of the isometric grid. This secondary pattern was frequently used as a single level design [Fig. 108a]. The use of the isometric grid as an underlying structure creates, by necessity, different thicknesses in the outer vertical and horizontal borders.

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Fig. 457

Figure 458a illustrates an exceptional type B dual-level design from the Topkapi Scroll.Footnote 81 While the background regions conform to a 3.6.3.6 arrangement of hexagons and triangles [Fig. 89], the widened lines of the primary design actually conform to the arrangement of hexagons, squares, and triangles in the 32.4.3.4-3.4.6.4 two-uniform tessellation [Fig. 90]. By selecting the triangles and squares to create the widened line, the artist was able to incorporate a secondary pattern that has an identical edge configuration in the underlying tessellations for these two repetitive elements. This was a very clever contrivance that more than makes up for the noncollinearity of the widened pattern lines associated with the neighboring primary hexagonal elements. Figure 458b illustrates the application of the underlying tessellation and the resulting acute pattern within the square and triangular repetitive cells. These place the dodecagons at each repetitive vertex and separate each with a barrel hexagon. The background regions in the primary design could have also been filled with the triangular repeat, but keeping these areas open creates the dual-level dynamic. The combined use of the square and triangular repetitive cells qualifies the secondary pattern as a hybrid design, and but for the arbitrary treatment of the pattern lines at the center of the triangular repeat unit, the hybrid use of these two repetitive cells is provided a second representation within the Topkapi Scroll, as a design suitable for an arch tympanum rather than as a dual-level design [Figs. 23d–f].Footnote 82 Figure 459a shows the application of the secondary underlying tessellation to the squares and triangles of the primary design. Of particular interest is the seamless incorporation of the rectilinear border in what is otherwise a 32.4.3.4-3.4.6.4 tessellation. Figure 459b applies the secondary median pattern lines to the underlying tessellation of Fig. 459a. Once again, the secondary pattern in this dual-level design is nonsystematic and works by virtue of the hybrid use of the square and triangular repetitive cells.

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Fig. 458

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Fig. 459

Figure 460a illustrates a Timurid type B dual-level design from the Gawhar Shad mausoleum in Mashhad (1416-18). Both the primary and secondary levels are median designs produced from the fourfold system A. Figure 460b shows the underlying tessellation that produces the secondary design along with the applied pattern lines of the median family. Much of the secondary design is the classic star and cross pattern created from the semi-regular 4.82 tessellation of squares and octagons [Fig. 124b]. Figure 461a shows the primary median pattern and its underlying generative tessellation. This is a field pattern comprised of superimposed octagons, and is one of several historical designs created from this simple underlying tessellation of large hexagons and squares [Fig. 138c]. Figure 461b demonstrates a method for widening the primary pattern lines that gives the proportions used in this historical example. Each edge of the superimposed octagons has eight applied squares, with the centers of the outer two squares placed upon the corners of the octagon. Figure 461c shows the widened line version of the primary design. Figure 461d places the secondary polygonal modules, along with their associated median pattern lines into the widened primary pattern. This pattern is only self-similar in its use of modules from the fourfold system A at two scales, and its employment of the median pattern family at both levels. However, the primary pattern has no eight-pointed stars, whereas the secondary pattern does; and the secondary pattern is not a widened line pattern. Therefore, this example is only loosely self-similar. The proportional scale between the two levels is 1:10.2426… which can be expressed as 6 + [√2 × 3].

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Fig. 460

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Fig. 461

Figure 462a represents an Aq QoyonluFootnote 83 cut-tile mosaic panel from the Friday Mosque at Isfahan (1475) [Photograph 99]. Figure 462b demonstrates how the widened primary design of this type B dual-level example is a simple assembly of squares and triangles that produces octagons, four-pointed stars, and concave octagons as background elements. Each triangle is a 1/8 segment of an octagon. Figure 462c details the application of the secondary polygonal modules to the primary squares and triangles, along with their associated median pattern lines. These modules are from the fourfold system A. The hybrid use of two distinct repetitive cells is similar in concept to the example in Figure 458. On their own, the square cells produce the ubiquitous star and cross pattern, while the triangular cells produce a very pleasing design that was used previously in several locations, including: the Haund Hatun complex in Kayseri (1237-38) [Fig. 157b]; the mausoleum of Uljaytu in Sultaniya, Iran (1305-1313) [Fig. 66]; and at the Bibi Khanum in Samarkand, Uzbekistan (1398-1404). Figure 462d shows the application of the secondary underlying polygonal modules and associated pattern lines to the complete panel.

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Fig. 462

Figure 463 represents a Qara Qoyunlu widened line dual-level design from the Imamzada Darb-i Imam in Isfahan (1453-54) [Photograph 98]. This design is created from the fivefold system, with the primary pattern being an unusual hybrid of acute and median widened pattern lines, and the secondary pattern being from the obtuse family. This is one of the most remarkable architectural examples of type B pattern making, and is the work of Sayyid Mahmud-i Naqash. (Note: The outer border is not represented.) Figure 464a illustrates the underlying tessellation along with the applied pattern lines for this example from the Darb-i Imam. The applied pattern lines are not collinear where they cross the midpoints of the underlying decagons. It is at these locations that the pattern lines change from acute to median. This is an unusual feature that ordinarily creates discontinuity within a given design, and only occasionally found within this ornamental tradition. In this widened line dual-level example, the noncollinearity is both eccentric and visually pleasing. Figure 464b shows how the proportions of the widened line are determined by the wide rhombus with 72° and 108° included angles. Figure 464c illustrates the application of the secondary underlying polygonal modules to the three regions that make up the widened line configuration in Fig. 464b. This also shows the secondary underlying polygons with the associated obtuse pattern lines. Figure 464d applies the three regions with secondary polygons and applied pattern lines to the widened lines of the primary pattern. The proportional scale factor between the primary and secondary levels is 1:14.3261… which can be expressed as 3 + [φ × 7], or as 4 + [φ × 5] + √5.

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Fig. 463

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Fig. 464

Figure 465 illustrates another type B dual-level design from the Topkapi Scroll.Footnote 84 This example uses the acute family in the primary level and the obtuse family within the widened lines of the secondary level. The patterns within both levels are produced from the fivefold system. Although this does not appear to have been used architecturally, this is arguably the most complex and successful type B dual-level design from the historical record. Figure 466a illustrates the primary pattern with its underlying generative tessellation. Figure 466b shows how the proportions within the widened lines of the primary pattern are determined by a series of rhombi, triangles, trapezoids and pentagon that all relate to the fivefold system. Figure 466c places the underlying polygonal modules onto the widened line segments, as well as shows the polygons with their associated obtuse pattern lines. Figure 466d shows the widened line primary pattern with the secondary generative polygons and associated pattern lines. As with many fivefold patterns, the median secondary pattern can also be constructed from an alternative grid with dual characteristics (not shown). The scale factor for this remarkable dual-level design is 1:17.9442… or as an expression of phi as 5 + [φ × 8], or as 9 + [√5 × 4].

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Fig. 465

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Fig. 466

3.3.3 Historical Examples of Type C Dual-Level Designs

Like type B dual-level designs, the type C design methodology involves the widening of the basic primary pattern, but extends the secondary pattern beyond the region of the widened lines so that it fills the entire plane. Differentiation between the primary and secondary patterns is provided in two ways: by emphasizing the widened lines themselves, as well as through color contrast between the region of the widened lines and its background. This is the least common variety of dual-level design with only a handful of examples known to the historical record. Figure 467a illustrates a Shaybanid type C dual-level design from a wooden ceiling at the Khwaja Zayn al-Din mosque and khanqah in Bukhara (c. 1500-1550). The widened line primary design is the classic threefold median pattern with six-pointed stars used repeatedly throughout the Islamic world [Fig. 95b]. Figure 467b shows how the proportions of the widened line are simply derived from the isometric grid (red). The secondary pattern repeats upon the isometric grid, and the differentiation between the primary and secondary levels in the wooden ceiling is achieved through relief rather than color. The secondary pattern places six-pointed stars at the vertices of the isometric grid. These are separated by hexagons located at the center of each triangular cell. Were it not for the inclusion of the widened lines within the overall pattern matrix the secondary pattern would function independently as ornamental surface coverage. This is a typical feature of type C dual-level designs.

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Fig. 467

Figure 468 represents one of the more complex historical type C dual-level designs. This Safavid example is from the Madar-i Shah in Isfahan (1706-14). Both levels are obtuse patterns created from the fivefold system, providing this design with a higher level of self-similarity than many historical dual-level designs. Figure 469a illustrates the underlying tessellation of edge-to-edge decagons and concave hexagonal interstice regions that is one of the most common generative tessellations within the fivefold system. The obtuse primary pattern lines are also represented. Figure 469b widens the obtuse primary pattern lines by using rhombi and triangles associated with fivefold proportions. This method of widening is very similar to that shown in Fig. 466b, although in this example the widening is applied to an obtuse rather than an acute pattern. Figure 469c places the secondary decagons onto the vertices of the widened primary pattern. The scale of the decagons is determined by applying the same arrangement of decagons and concave hexagons to the rhombic regions as is used in the primary underlying tessellation wherein the four decagons surround a concave hexagon. This feature, and the fact that both the primary and secondary patterns are of the same pattern family, provides scale invariance between the small rhombic regions of the secondary pattern and rhombic repeat of the primary design. Figure 469c also shows the full infill of the secondary polygonal modules. The additional decagons within the primary ten-pointed stars are located at the vertices of the array of rhombi located within each of the primary ten-pointed stars in Fig. 469b. Figure 469d applies the associated obtuse pattern lines to the secondary underlying polygons. It is worth noting that both the primary and secondary obtuse patterns can be derived with equal ease from an alternative underlying tessellation with dual characteristics comprised of decagons, pentagons, barrel hexagons and thin rhombi [Fig. 200]. The scale factor between the primary and secondary levels is 1:15.3262… or as an expression of phi as 4 + [φ × 7], or as 6 + [φ × 3] + [√5 × 2].

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Fig. 468

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Fig. 469

3.3.4 Historical Examples of Type D Dual-Level Designs

Type D dual-level designs are similar to the type C variety in that the secondary pattern has complete surface coverage and the primary pattern is expressed as a widened line. However, the visual characteristics of these two types of dual-level design are very distinct from one another. Type D designs rely upon the stars of the secondary pattern having parallel lines that have a collinear orientation with those of neighboring stars. This creates a channel of secondary background elements that are provided their own color, thereby differentiating the widened line from the rest of the design. Without this color, the design would appear as a standard, albeit rather complex, geometric design. Figure 470a represents a Mudéjar type D dual-level design from the Alcazar in Seville (1364-66) [Photograph 101]. Like the majority of dual-level designs from the Maghreb, both the primary and secondary patterns are associated with the fourfold system A. To demonstrate the reliance upon color to differentiate the primary pattern in this variety of dual-level design, the geometric pattern in Fig. 470b is the secondary design alone. As is visually apparent, it is next to impossible to ascertain the primary design within this overall pattern matrix. Figure 471a illustrates the underlying tessellation along with the associated pattern lines of the obtuse family. Figure 471b places secondary octagons at the intersections of the primary pattern. These are scaled such that a ring of eight edge-to-edge octagons fits onto each primary eight-pointed star. This illustration demonstrates how the lines of each small secondary eight-pointed star are parallel to their immediately adjacent lines of the primary pattern. Figure 471c fills in the background regions from Fig. 471b with additional polygonal modules from the fourfold system A. Figure 471d shows the primary pattern as single lines, more or less in the style of type A dual-level designs. The proportional scale factor between the primary and secondary levels is 1:4.8284… which can also be expressed as 2 + (√2 × 2).

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Fig. 470

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Fig. 471

Figure 472 represents a Nasrid zillij mosaic dual-level panel from the Alhambra [Photograph 102]. The primary and secondary patterns are also created from the fourfold system A. Figure 473a shows the derivation of the primary acute pattern from an underlying tessellation of octagons, pentagons and triangles. This pattern repeats upon a rectangular grid. Figure 473b places octagons at the intersections of the primary pattern. As with the example from the Alcazar in Fig. 470, the size of the octagons is determined by their eightfold edge-to-edge placement around the primary eight-pointed stars. The associated pattern lines are also from the acute family, giving this dual-level design a strong self-similar characteristic. Figure 473c places further octagons (blue) at key locations within the primary pattern. The locations of this new set of octagons are determined by the underlying tessellation (red lines) of the primary design. Figure 473d fills in the remaining background with additional polygonal modules from the fourfold system A. And like the example from the Alcazar, the proportional scale factor between the primary and secondary levels is 2 + (√2 × 2), or 1:4.82842712.

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Fig. 472

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Fig. 473

Figure 474 represents a Marinid type D dual-level design from the Bu’Inaniyya madrasa in Fez (1350-55). This is one of the few dual-level designs from the Maghreb that is created from the fivefold system. Rather than a conventional geometric pattern, the primary design is a polygonal tessellation of decagons that are placed edge to edge in the horizontal orientation, and corner to corner in the vertical orientation, and separated by eight-sided interstice regions. The secondary pattern is from the acute family, and the discrepancy between the visual quality and methodological origin between the primary and secondary patterns means that this dual-level design does not possess self-similarity. It is nevertheless a very appealing design. The arrangement of decagons in the primary pattern produces an overall repeat unit that is rectangular. Figure 475a illustrates the aforementioned arrangements of decagons touching edge-to-edge horizontally, and corner-to-corner vertically. Figure 475b subdivides this decagonal arrangement into smaller polygonal components. Figure 475c applies scaled-down polygonal modules from the fivefold system, along with their associated acute pattern lines, to the four varieties of polygonal component from Fig. 475b. Each of these four polygonal components can be used to created patterns on their own, and qualifies this secondary pattern as a hybrid design, much like the group of repetitive fivefold elements in several Anatolian Seljuk hybrid patterns [Figs. 263–265]. Figure 475d applies the secondary elements and associated pattern lines from Fig. 475c to the overall structure in Fig. 475b. The scale factor between the primary decagons in Fig. 475a and the secondary underlying decagons in Fig. 475d (blue) is 1: 4.2360… or as 1 + [φ × 2], or as 2 + √5.

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Fig. 474

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Fig. 475

Figure 476 shows another Marinid design that makes use of the same arrangement of decagons as the example in Fig. 474. This design is from the al-‘Attarin madrasa in Fez (1323) [Photograph 103], and, like the previous example, the secondary level is an acute pattern created from the fivefold system. The obvious difference between these two Marinid dual-level designs is the level of complexity of the secondary pattern. As with the dual-level design at the Darb-i Imam [Figs. 451 and 452], and the highly complex pattern that shrouds the Gunbad-i Qabud in Maragha [Figs. 239 and 240], it has been argued that this dual-level design from the al-‘Attarin madrasa is quasicrystalline.Footnote 85 While the isolated region of local tenfold symmetry contained within each primary decagon undoubtedly has shared characteristics with decagonal quasicrystals, the fact remains that this decagonal region is only a subset of design elements that exists as part of a larger pattern matrix that employs a rectangular repeat with translation symmetry—the antithesis of quasicrystallinity. Figure 477a illustrates the origin of the ten-pointed stars at the centers of each primary decagon. Figure 477b divides the structure into smaller repetitive components that are essentially the same as the previous example from the Bu’Inaniyya madrasa. Figure 477c places modules from the fivefold system into the construction from Fig. 477b. The scale of these secondary modules is determined by an arrangement of three linear decagons separated by two sets of mirrored pentagons, with the distance between the centers of the outer two decagons equaling the edge length of the primary decagons. Figure 477d places acute pattern lines into the secondary polygonal modules. The similarities between the previous example from the Bu’Inaniyya madrasa and this example from the al-‘Attarin madrasa, and their closeness in proximity and date suggest the strong possibility that they were designed by the same individual or are the product of the same artistic lineage. The scale factor between the primary decagons in Fig. 477a and the secondary underlying decagons in Fig. 477c (blue) is precisely double that of the previous example, which is to say: 1:8.4721… or as 2 + [φ × 4], or as 4 + [√5 × 2].

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Fig. 476

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Fig. 477

3.3.5 Potential for New Multilevel Designs

In both the east and west, the number of overall examples of dual-level designs is relatively small. This is somewhat surprising considering their visual and intellectual distinction. This relative rarity is presumably due to the highly specialized design methodology required of this tradition, and a consequent paucity of specialists familiar with this dual-level discipline. It is certainly not the case that this rarity is in any way the result of an exhaustion of the creative potential that this methodology offers. On the contrary, multilevel design methodology offers contemporary artists an unlimited capacity for highly innovative original designs that expand upon the remarkable examples of the past. As detailed above, historical dual-level designs were of just four varieties, and utilized just four of the design systems: the system of regular polygons, the fourfold system A, the fourfold system B, and the fivefold system. In addition, all historical examples repeat with conventional translation symmetry. Even constraining oneself to these features, there is limitless potential for the creation of designs with great beauty and originality. However, contemporary explorations in multilevel design allow for significant innovation beyond these methodological constraints, including: new varieties of design beyond the types A, B, C, and D; examples that employ more than just two recursive levels of design; multilevel designs that employ the sevenfold system; self-similar designs with fully realized scale invariance; and aperiodic multilevel designs that are truly quasicrystalline. Undoubtedly, this is an area of design that has extraordinary potential to contemporary geometric artists.

As shown, the primary patterns in the two historical fivefold type D examples from Fez are a grid of decagons rather than a conventional geometric pattern. By contrast, as per Figs. 470 and 472, artists in the Maghreb typically used conventional geometric patterns for the primary pattern in their fourfold type D dual-level constructions. Although the fivefold system was not used in this way historically, it will make very acceptable type D dual-level designs. The design in Fig. 478 (by author) uses the classic fivefold obtuse pattern for the primary level, and an acute pattern for the secondary level. Figure 479a shows how the secondary pattern is created from the placement of underlying generative decagons at the intersections in the primary pattern lines, as well as at the center of the primary ten-pointed star. The size of these secondary decagons is determined by an arrangement of two decagons separated by mirrored pentagons, with the distance between the centers of the decagons being equal to the edge of the pentagons within the primary pattern. Figure 479b adds the secondary acute pattern lines to this underlying secondary tessellation. The proportional scale between the two levels is 1:5.2360… and as a function of phi can be expressed as 2 + [φ × 2], or as 3 + √5. This is the same scale factor as one of the dual-level designs in the Topkapi Scroll [Fig. 449].

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Fig. 478

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Fig. 479

As mentioned, though no historical examples are known, the modules that comprise the fivefold system can be used to create patterns with five- and tenfold local symmetry that fulfill the modern mathematical criteria for aperiodic quasicrystallinity with scale-invariant self-similarity. Several contemporary artists and designers working with Islamic geometric design, including the author, are exploring the application of this ancient design discipline to these areas of mathematical inquiry.Footnote 86 While such innovation is beyond the scope of this work, the small selection of examples that follow reveal the remarkable similitude between these areas of modern mathematical discovery and historical design systems developed by Muslim artists a thousand years previously.

The median pattern in Fig. 480 (by author) has no translation symmetry. This is an aperiodic design that employs the 2 fivefold rhombi with the edge matching rules discovered by Sir Roger Penrose in the 1970s: one with 36° and 144° included angles, and the other with 72° and 108° included angles. In this instance, the edge matching rules are manifest in the geometric pattern matrix that is applied to each variety of rhombus. As demonstrated previously, Muslim artists used both of these rhombi historically as repeat units. However, by synchronizing the pattern lines upon the relevant edges of each rhombus so that they conform to Penrose matching rules, the fivefold designs created from tessellating with these two rhombi have forced aperiodicity. While this example is aperiodic, it only has a single level of design. However, just as the two Penrose rhombi are well known for their ability to infinitely inflate and deflate, so also can the geometric patterns that are applied to these rhombi be provided with inflation and deflation. Such designs have both scale-invariant self-similarity and true quasicrystallinity. Figure 481 shows an example of a three-level quasicrystalline design (by author) created from the fivefold system. As with the many historical dual-level designs created from the fivefold system, there are multiple regions with tenfold local rotation symmetry in each level. However, the overall structure of this example is aperiodic, and this same aperiodic pattern is recursively replicated at each successive level, providing this design with scale invariant self-similarity. The pattern in any given level will inflate and deflate infinitely, and the scaling ratio is phi, or 1:1.6180…. The aesthetic treatment of this example is not at all traditional, and employs transparency in the overlay of the three levels. This visually emphasizes a feature of particular note in this geometric multilevel geometric structure: the frequency at which the intersection points of one level align with key positions in the pattern lines of another level. Figure 482 illustrates the two rhombic repetitive cells that produce the aperiodic designs in Figs. 480 and 481. As determined by both the underlying generative tessellation and the resulting geometric design, the edges of these two rhombi are constrained by Penrose matching rules. The applied geometric pattern lines within each rhombus are governed by lines of reflected symmetry (as per the black dashed lines). These identify Penrose’s inflationary and deflationary subdivisions. The Inflation and deflation of the geometric design in Fig. 481 is achieved through recursively applying scaled versions of these two rhombi into the sequential subdivisions at each level; thereby insuring that each level is aperiodic and provided with scale invariant self-similarity. This process of the recursive application of scaled down modules is controlled by the Penrose subdivision rule, and is sometimes referred to as substitution tiling. Figure 483 illustrates this process as applied to the Penrose rhombi. As they inflate and deflate to each successive level the matching rules are maintained, providing each recursive level with self-similar aperiodicity. The inflation and deflation ratio is phi, or 1:1.6180…. The three-level design in Fig. 481 applies the geometric designs for both the rhombi in Fig. 482 to each successive level of recursive subdivision. The recursive application of these two decorated rhombi infinitely fills the two-dimensional plane aperiodically, and infinitely inflates and deflates.

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Fig. 480

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Fig. 481

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Fig. 482

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Fig. 483

The fivefold system can also be used to create self-similar designs and quasicrystalline designs with radial symmetry. The example in Fig. 484 (by author) is a two-point design with tenfold rotational symmetry (only 1/4 shown) that has the added feature of diminishing in scale as the design moves outward from the center. Except for their scale, the primary design (black) and the secondary design (red) are exactly the same, providing this self-similar design with scale invariance. The proportional scale between the two levels is 1 + φ, or 1:2.6180…. As with all scale invariant self-similar designs, this same design has the ability to be recursively applied infinitely. Figure 485 demonstrates how the outward diminution of the primary and secondary patterns in Fig. 484 results from the use of underlying polygonal modules from the fivefold system that sequentially reduce in scale as they move from the center to the periphery. In this illustration the outward expansion has been stopped arbitrarily, but there is no limit to this expansion: all the while with the tessellating polygons becoming smaller and smaller. In fact, the outward expansion of this self-similar structure is an example of Zeno’s paradox whereby the ongoing diminishing expansion infinitely approaches, but never arrives at a theoretical limit. Figures 269 and 270 illustrate two of the exceedingly rare historical fivefold patterns that make use of underlying generative tessellations comprised of systematic modules that have more than a single scale. However, the variable scales in these historical examples phase back and forth between just two scales. They do not diminish infinitely outward, nor are they dual level. As mentioned previously, there are two edge lengths in the polygonal modules of the fivefold system. The diminishing scale of both the primary and secondary underlying tessellations in this design is the result of scaling down the polygons so that their long polygonal edges match the shorter edges of the un-scaled set of polygons, thereby sequentially reducing size as they move outward. Whereas the ratio between the recursive application of the primary and secondary levels is 1 + φ, the ratio between each sequential reduction of the polygonal modules within each tessellation is just phi.

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Fig. 484

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Fig. 485

The dual-level design in Fig. 486 (by author) is created from the sevenfold system. This repeats upon a rhombic grid and is self-similar in that the 14-s4 median pattern family is used at both scales, and the application of this variety of pattern can be recursively applied to each new level infinitely. However, while there are regions of similitude between the levels, this design does not have scale invariance. Also, this design has translation symmetry by virtue of its rhombic repeat unit, and while self-similar, is not aperiodic. Figure 487 illustrates the method for created the design from Fig. 486. This follows the procedure used in historical examples of dual-level designs wherein scaled-down primary polygons—in this case tetradecagons—are applied to the vertices of the primary pattern. As with dual-level designs created from other generative systems, the size of the scaled-down tetradecagons is determined by their application to the vertices of the primary pattern. Figure 487a shows three edge configurations: one where tetradecagons are meeting edge-to-edge; one where they overlap and intersect at their corners; and one where they are separated by the convex hexagons from the sevenfold system. Figure 487b demonstrates how these scaled tetradecagons are placed on the vertices throughout the primary pattern. Figure 487c fills in the edge lengths of the primary pattern with further polygonal modules from the sevenfold system; and Fig. 487d shows the completed application of secondary modules to the total design. Rho [ρ] and delta [σ] being the two proportions inherent to the heptagon [Fig. 277], the scaling ratio between the primary and secondary levels is 1:8.0978… which can be expressed as [ρ + σ] × 2.

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Fig. 486

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Fig. 487

The three-level example in Fig. 488 (by author) is also created from the sevenfold system. This design is quasicrystalline, with 14-fold rotational symmetry (only 1/4 shown), and its self-similarity is scale invariant. As an alternative to standard historical dual-level methodology, this design places scaled-down 14-pointed stars on the vertices of the underlying tessellation rather than at the vertices of the geometric pattern. Each of the three levels of this design uses the crossing pattern lines of the 14-s4 median family [Fig. 272]. The recursive iterations of this pattern are scaled down from the center of the design (upper left corner). The primary pattern is blue, the first recursion is green, and the second brown. Each of these is in a tiling treatment, and provided with a level of transparency. Needless to say, such transparency is not a feature of historical practices. Each subsequent iteration can be extended outward from the center infinitely. The scaling ratio between each iteration is 1 + ρ + σ, or 1:5.0489…. Figure 489 illustrates the three underlying generative polygonal structures for the sevenfold quasicrystalline design from Fig. 488. The first level of pattern is created from the large bounding tetradecagon (black) in Fig. 489a. The second level of pattern is created from the polygonal infill of this bounding tetradecagon in the same illustration. The scale of the secondary tessellation is determined by placing tetradecagons at the vertices of the primary tetradecagon that are separated by the concave hexagons associated with this system [Fig. 271]. A further secondary tetradecagon of the same size is strategically placed at the center of the primary tetradecagon. Figure 489b scales down the polygonal modules in the secondary tessellation in Fig. 489a so that the same configuration fits within the small secondary tetradecagon at the center of the primary tetradecagon. This scaled-down assembly of polygons is also applied within the secondary (partial) tetradecagons located at the vertices of the primary tetradecagon. Small third-level tetradecagons are then placed at each vertex of the secondary tessellation, and further infill with modules from the sevenfold system completes the third level of generative tessellation. This scale invariant self-similar process can be recursively iterated infinitely, always with the same scaling ratio of 1 + ρ + σ, producing a quasicrystal with 14-fold rotational symmetry.

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Fig. 488

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Fig. 489

3.4 Geometric Ornament on Domes: Radial Gore Segments

There are two historical repetitive stratagems for applying geometric designs onto the surfaces of domical structures: polyhedral symmetries, and radial gore segments. Both of these domical methodologies were pioneered by the Seljuks, and both were employed by subsequent Muslim cultures for applying geometric designs to the interior and exterior surfaces of domes, as well as to the quarter dome hoods of mihrab niches. Both of these repetitive methodologies lend themselves to the three-dimensional application of the polygonal technique, and both are aesthetically successful, albeit visually distinct from one another. However, while a large number of examples that employ radial gore segments are found throughout Muslim cultures, only a relatively small number of polyhedral examples are known to the historical record.

The use of radial gore segments in applying geometric designs onto the surfaces of domical structures has the advantage that it will work equally well with both hemispherical domes and domes with a pointed apex. There are two convenient methods of deriving profile curvatures for this latter category of dome: the use of key points associated with either the orthogonal grid or individual polygons. These will sometimes have a single center point for each side of the profile, but are more frequently created from two points of curvature for each side.Footnote 87 The gore segmentation of the surface of domes historically favored 8-, 12-, and occasionally 16-fold radial divisions, although other divisions, such as 6-fold and even 24-fold, are also known. As a rule, these divisions adhere to the symmetry of the supporting chamber, and with the vast majority of domes being supported by structures with a square floor plan, the radial divisions are almost always multiples of four. Figure 490 illustrates the eightfold radial segmentation of the hemisphere. Figure 490a shows a single gore segment laid flat upon the two-dimensional plane, Fig. 490b shows the dome in elevation, Fig. 490c shows the dome in plan, and Fig. 490d shows an array of the eight radial segments laid flat. Of course a dome is a double-curved surface and will not unfurl onto the two-dimensional plane without distortion. The following illustrations of historical geometric gore segments are therefore only representational of the actual geometry, but nonetheless demonstrate the prevalent use of the polygonal technique in laying out geometric designs on gore segments with widely diverse proportions.

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Fig. 490

The earliest extant dome that is ornamented with a geometric design based upon radial gore segments is from the Friday Mosque at Gulpayegan (1105-18). Figure 491a illustrates the underlying polygonal tessellation that creates the design on this Seljuk dome. The ring of eight edge-to-edge heptagons that create the ring of seven-pointed stars illustrate a fundamental principle in the placement of geometric designs upon gore segments: the precise curvature of the dome and the underlying generative tessellation are intrinsic to one another. Figure 491b represents the unfurled eightfold segments. At the outer corners of each segment are 1/4 portions of eight-pointed stars. These produce the 8 half eight-pointed stars located at the base of this dome. It is interesting to note the similarity between the 2 ten-sided motifs directly below, and one immediately above the seven-pointed stars and those found in two of the patterns from the northeast dome chamber in the Friday Mosque at Isfahan [Figs. 261 and 309b]. This was evidently a popular devise among Seljuk artists during the turn of the twelfth century.

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Fig. 491

Figures 492 and 493 illustrate the use of the polygonal technique for creating the geometric designs of four Mamluk domes based upon radial gore segmentation.Footnote 88 The first three of these four examples are from the notable group of Cairene domes with exterior monochromatic carved stone that achieves design clarity through high relief and resulting shadow. As with the previous Seljuk example from Gulpayegan, the precise curvature and proportions of the gore segments for each of these examples supports the arrangement of underlying polygons that produce the geometric designs. Figure 492a represents the 1/20 gore segment of the dome covering Sultan Barsbay’s mausoleum at the Sultan al-Ashraf Barsbay funerary complex at the northern cemetery in Cairo (1432-33) [Photograph 61]. This dome places sequential rings of 20 half eight-pointed stars at the base, followed by full eight-pointed stars. The pattern in this lower section of the dome is a median design with 60° angular openings applied to the 4.82 tessellation of octagons and squares [Fig. 126a]. Immediately above these eight-pointed stars is a ring of seven-pointed stars, and above this is a region of interweaving hexagons and triangles that can be described as a tapered form of the well-known median pattern, albeit with pattern lines that continue through the otherwise six-pointed stars. This design is easily created from the 63 grid of regular hexagons [Fig. 95b]. This section is surmounted by a ring of distorted 7-pointed stars, with a 20-pointed star the apex of the dome that is only visible in toto from a bird’s eye view. The bold interweaving widened lines that comprise this example are represented in Fig. 493a. The gore segment in Fig. 492b is from a dome at the same Mamluk funerary complex, and covers the tomb of an anonymous Barsbay family member. The radial gore of this domical geometric design employs a sixfold segmentation of the dome. The primary 8- and 12-pointed stars are derived from a tessellation of underlying dodecagons and octagons in the lower 3/4 of the design. It is interesting to note that two-dimensional version of the domical design produced from these underlying dodecagons and octagons was used by Mamluk artists just 10 or 15 years previous at the Sultan al-Mu’ayyad Shaykh complex in Cairo (1415-22) [Fig. 380e]. The design in this lower region transitions into sequential bands of 5-pointed stars, culminating in a single 12-pointed star at the apex of the dome. Figure 493b illustrates the interweaving treatment of this design. The design in Fig. 492c shows the geometric component from the dome of the Sultan Qaytbay funerary complex in the northern cemetery in Cairo (1472-1474) [Photograph 2]. This exceptional dome incorporates a floral motif (not shown) that meanders beneath the geometric design. The ornament of this dome repeats upon a eightfold radial segmentation, and each gore segment places a ring of half 10-pointed stars at the base of the dome, followed by a ring of 9-pointed stars, followed by a ring of rather distorted 5-pointed stars, and culminating in a 16-pointed star at the apex. Figure 493c illustrates the widened geometric lines from the 1/8 gore segment of this historical example. Figure 492d shows a design from the mosaic mihrab hood at the Amir Qijmas al-Ishaqi mosque in Cairo (1479-81). This is based upon an sixfold radial segmentation, with half 12-pointed stars at the base of the semi-dome, followed by 10-pointed stars at the middle of each radial gore, and half a 12-pointed star at the apex of the dome. Figure 493d illustrates the widened line treatment of the geometric ornament of this example. The very different proportions of this 1/6 gore segment and the previous 1/6 gore segment from the anonymous Barsbay family member’s tomb in Fig. 492b results from the difference in the dome profiles of these two examples. The example from the Barsbay family member’s tomb is very high in relation to its diameter, whereas the example from the Amir Qijmas al-Ishaqi mosque is much lower in relation to the diameter of its base. This results in the differences between the height and width proportion of these two examples.

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Fig. 492

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Fig. 493

The four gore segments with applied geometric patterns in Figs. 494 and 495 are from Persia and Central Asia. The ornament of each of these four domes is polychromatic cut-tile mosaic. Figure 494 demonstrates the application of the polygonal technique to domical gore segments to generate these designs. The design in Fig. 494a is from the interior of a Muzaffarid dome in the Friday Mosque at Yazd (1324) [Photograph 91]. This design repeats upon a 16-fold radial segmentation, and places half 6-pointed stars at the base, followed by 7-pointed stars, more 6-pointed stars, 5-pointed stars, 4-pointed stars, and a 16-pointed star at the apex. Figure 495a provides a widened line version with representative color differentiation within the background regions. The design in Fig. 494b is from the interior dome of the mausoleum of Turabek-Khanym in Konye-Urgench, Turkmenistan (1370) [Photograph 92]. This was produced during the short-lived reign of the Sufi Dynasty, and is an aesthetic precursor to the remarkable architectural ceramics of the Timurids. This dome is divided into 12 radial gore segments, with half ten-pointed stars at the base, followed by ten-pointed stars, and nine-pointed stars. Between these primary stars is a connective pattern matrix typical of the obtuse family. This design transitions into a 24-pointed star at the apex. Figure 495b provides the widened line version of this design as per the historical example, but without the highly ornate floral background mosaics. Figure 494c is the 1/8 gore segment from the Safavid exterior dome of the Friday Mosque at Saveh (late sixteenth century). The underlying tessellation for this design employs half dodecagons at the base of the dome that starts an ascending progressive sequence of octagons, hendecagons, and nonagons that are separated by elongated hexagons and concave hexagons. These produce an obtuse pattern that places half 12-pointed stars at the base of the dome, followed by 8-pointed stars, 11-pointed stars, 9-pointed stars, and an 8-pointed star at the apex. Figure 495c represents the widened line version of this design that was used in the historical example. It is worth mentioning that a very similar geometric pattern is also used on the interior of the dome in Saveh (not shown). This interior dome places half ten-pointed stars at the base, followed by nine-pointed stars, ten-pointed stars, seven-pointed stars, and culminating with an eight-pointed star at the apex. The combined use of sequential numbered stars (seven-, eight-, nine- and ten-pointed) is an artistic device that was also used in many two-dimensional designs—especially by artists working during the Seljuk Sultanate of Rum. The design in Fig. 494d is from one of the most well-known exterior geometric domes in Iran: the Safavid shrine of Aramgah-i Ni’mat Allah Vali in Mahan (1601) [Photograph 93]. The design on this dome is regulated by a 12-fold radial segmentation. The base of the dome has half 8-pointed stars, followed by 10-pointed stars, 9-pointed stars, 11-pointed stars, 12-pointed stars, 9-pointed stars, 7-pointed stars, 5-pointed stars, and culminating in a 12-pointed star at the apex. Here again the artist strove to construct a design with a consecutive numeric sequence (7-, 8-, 9-, 10-, 11-, and 12-pointed stars). This achievement required considerable skill, even if some of the stars are relatively irregular, and not placed in ascending numeric order. Figure 495d provides a representation of the widened line treatment of the pattern on this dome.

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Fig. 494

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Fig. 495

3.5 Geometric Ornament on Domes: Platonic and Archimedean Polyhedra

The most geometrically interesting and visually arresting non-Euclidean Islamic geometric designs are the very rare examples that employ polyhedral symmetry as their repetitive schema. In the case of applying patterns to domical gore segments, other than having to balance the underlying generative tessellation with the tapering curvature of the n-fold segment, the basic geometric constraints are more or less the same as those governing two-dimensional pattern making. By contrast, applying geometric designs onto the surface of the sphere involves geometric conditions that have no parallel on the two-dimensional plane. This creates interesting geometric challenges and aesthetic opportunities for that artist, and it is surprising that there are so few historical examples of this form of domical ornament.

The earliest known example of polyhedral geometric ornament is from the interior of the magnificent northeast dome in the Friday Mosque at Isfahan (1088-89) [Photograph 30]. This geometric design is created from the pentagonal faces of the dodecahedron projected onto the curvature or the domical surface. Each of these pentagonal faces is used as an underlying generative module in the same way that pentagons are used to create fivefold patterns on the two-dimensional plane, the difference being that on the sphere pentagonal faces can tessellate on their own, whereas on the plane they require at least one other module from the fivefold system for complete surface coverage. Figure 496 illustrates the two-point pattern constructed from the spherical projection of the dodecahedron. This follows the conventions of the polygonal technique by using the projected pentagonal faces (orange lines) as the underlying generative tessellation. This example from Isfahan centers the design at the apex of the dome on one of the pentagonal background elements; thereby creating a fivefold rotational symmetry that descends from the apex to the base of the dome. This is at odds with the ascending 4-, 8-, and 16-fold symmetry within the cubical supporting structure for this dome. One would expect this dome to be a hemisphere that would correspond with half of the spherically projected dodecahedron. However, this remarkable dome is provided with additional loft wherein the apex is raised above the surface of the hemisphere. The artist’s decision to break the symmetry of the dodecahedron provides the domed chamber with greater spatial volume, introducing a central point for the eye to fix upon. Considering that this is a mosque, one can assume that this may have also been intended as a way of spatially emphasizing a concept of religious ascendancy. The application of this geometric design therefore required the projection of the original pentagonal faces of the dodecahedron beyond the hemispherical surface onto the elevated surface. The resulting distortion is minimal, and in no way detracts from the beauty of this dome. In fact, this innovative change to the profile of the dome augments the beauty of the geometric design. A more conventional example of this same non-Euclidean design was used on a projecting hemispherical stone detail in the arch spandrel of the entry portal at the Sahib Ata mosque in Konya (1258). Other than the fact that this example is hemispherical, the main difference between this example from the Seljuk Sultanate of Rum and the earlier example from Isfahan is that the pattern lines of the later example are given a curvilinear treatment. This creates fivefold floriated elements within the regions that are otherwise pentagons.

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Fig. 496

Another projecting carved stone hemisphere from the Seljuk Sultanate of Rum also makes use of the dodecahedron as the repetitive schema for its geometric design. This is found in the Huand Hatun complex in Kayseri (1237), and like the example from the Sahib Ata mosque, this example is also located within an arch spandrel, although this example is in a mihrab rather than an entry portal. Figure 497 illustrates a full spherical representation of the median pattern that was used for the high relief stone ornament of the projected hemisphere at the Huand Hatun complex. Like the example from Isfahan, this makes use of the projected pentagonal faces as the underlying tessellation upon which the median pattern lines are applied. This pattern is characterized by a simple spherical matrix of five-pointed stars located within each underlying pentagon, and ditrigonal hexagons with threefold symmetry located at each vertex of the underlying tessellation.

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Fig. 497

The previous domical examples are derived from the dodecahedron. Each of the five Platonic solids has only a single type of polygonal face, and a single variety of vertex condition: the tetrahedron has four triangular faces and four 33 vertices; the cube has six square faces and eight 43 vertices; the octahedron has eight triangular faces and six 34 vertices; the dodecahedron has twelve pentagonal faces and twenty 53 vertices; and the icosahedron has twenty triangular faces and twelve 35 vertices. Of the five Platonic solids, only the dodecahedron appears to have been used in a fashion whereby the faces of the polyhedra are treated as an underlying generative tessellation. However, out of the 13 Archimedean solids, at least two historical domical examples make use of the projected spherical faces of the polyhedra as underlying tessellations. Archimedean solids are characterized by two or more varieties of regular polygonal face and identical vertices. If the Platonic solids are analogous to the two-dimensional regular grids, the Archimedean solids are analogous to the two-dimensional semi-regular grids [Fig. 89]. Other than the dodecahedron and icosahedron, the use of the Platonic solids as underlying generative tessellations would produce very simplistic patterns, and it is perhaps not surprising that these polyhedra were not used to create spherical designs. However, the greater complexity of the Archimedean solids would have afforded artists significant design potential, and it is surprising that only two of these polyhedra appear to have been used historically as underlying generative tessellations.

Figure 498 illustrates a median design applied to a spherical projection of the truncated icosahedron. This Archimedean solid is made up of 12 pentagonal and 20 hexagonal faces, and 60 identical vertices with a 5.62 arrangement of pentagons and hexagons. This is commonly recognized as the standard soccer ball. This polyhedra was used to create a geometric design in the hood of the mihrab arch at the Lower Maqam Ibrahim in the citadel of Aleppo (1168). This masterpiece of Zangid woodwork is signed by Ma’ali ibn Salam. The pattern lines that make up this spherical design are a combination of great curves and offset great curves. The double curvature of the offset great curves applies to the parallel lines that connect neighboring five-pointed stars. These would have been particularly complex to calculate and construct accurately in wood, and the precision of this spherical woodwork is testament to the genius of this artist. This same spherical design was used many centuries later for one of the purely ornamental Ottoman hollow pierced wooden balls that hang from the ceiling in the mausoleum of Mevlana Jalal al-Din al-Rumi in Konya. The only difference between the design of the earlier Zangid example from Aleppo and that found in Konya is that the six-pointed stars located within the underlying hexagons are open stars in the Ottoman wooden sphere. By contrast, the lines of the six-pointed stars in the Zangid example run through their underlying hexagon, thereby creating a central hexagon surrounded by six triangles.

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Fig. 498

The spherical design in Fig. 499 can be produced from the truncated cube. This example is from the Seljuk Sultanate of Rum, and is found in another projecting stone hemisphere in the arch spandrel of the portal at the Susuz Han in Susuzköy, Turkey (1246). The truncated cube has 14 faces, 8 being triangular and 6 being octagonal. The 24 identical vertices are in a 3.82 arrangement. The pattern lines place an eight-pointed star rosette within each underlying octagon. These eight-pointed stars are located upon the vertices of the octahedron. The proportion and placement of the parallel lines of the eight-pointed stars are determined by their point of origin being the midpoints of the adjacent underlying triangles, thereby creating a hexagon within each underlying triangle. It is worth noting that this pattern can also be created from an underlying polygonal network made up of octagons connected with trapezoids and triangles that are located within the eight triangular faces of the octahedron. The generative schema for this spherical design has two analogous two-dimensional examples from Turkey that are roughly of the same time period. The same application of pattern lines was used in the two-dimensional triangular repetitive cells from a panel in the Mengujekid minbar at the Great Mosque of Divrigi (1228-29), and in the carved stone ornament at the Çifte Minare madrasa in Sivas, Turkey (1271). Other than the difference in spherical versus planar topology, these examples differ in the type of primary stars located at the vertices where the triangular elements come together. In the case of the octahedron, the eight-pointed stars result from there being two points at each corner of the triangular cell, and four projected triangular faces meeting at each vertex. By contrast, in the case of the two-dimensional design, the vertices have six triangular repetitive cells, and the two points within each corner of the triangle produces the 12-pointed stars [Fig. 320]. The fact that both this two-dimensional design and its analogous spherical variant were being used in the same region and during the same period of time would appear to be more than coincidental.

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Fig. 499

As referenced in the previous example, while spherical projections of the less complex Platonic solids were not particularly suitable for use as underlying tessellations, they were occasionally used as repetitive devices upon which more complex nonsystematic underlying tessellations could be constructed. Figure 500 illustrates a spherical design that uses the octahedron (black lines) as its governing repetitive structure. This design was used in the Ayyubid mihrab hood of the al-Sharafiyah madrasa in Aleppo (before 1205). This is produced in low relief carved stone and is signed by ‘Abd al-Salâh Abû Bakr. The underlying generative tessellation for this design places octagons at the vertices of the octahedron, and surrounds each of these with a ring of eight pentagons (red lines). Like the spherical design from the Susuz Han, the triangular repeats of this Ayyubid example also have a two-dimensional analogue, and like the previous example, the 8-pointed stars at the vertices of the octahedron become 12-pointed stars on the two-dimensional plane [Fig. 300a—acute].

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Fig. 500

The spherical design in Fig. 501 is also created from a nonsystematic underlying tessellation that repeats upon the projected triangular faces of the octahedron. This acute design was used by the Ayyubid artist Abu al-Husayn bin Muhammad al-Harrani ‘Abd Allah bin Ahmed al-Najjar in the hood of the wooden mihrab (1245-46) of the Halawiyya mosque and madrasa in Aleppo. This design places underlying octagons at the vertices of the octahedron (black lines), nonagons at the center of each triangular face of the octahedron, and surrounds these with connecting pentagons and barrel hexagons (red lines). Once again, this design is analogous to a well-known two-dimensional pattern that places 12-pointed stars at the vertices of the isometric grid and 9-pointed stars at the centers of each repetitive triangle [Fig. 346a]. The octahedral form of this design transforms the 12-pointed stars into 8-pointed stars, creating an immensely successful spherical pattern comprised of 8- and 9-pointed stars connected by a pattern matrix of 5-pointed stars and opposing darts.

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Fig. 501

The spherical two-point design in Fig. 502 is arguably the most complex historical example of an Islamic geometric pattern based on polyhedral geometry. This non-Euclidean design is found in a mosaic arched hood at an anonymous Mamluk mausoleum in the Nouri district of Tripoli, Lebanon. The repetitive structure of this design is the cuboctahedron, comprised of eight triangular faces and six square faces. The 12 identical vertices are in a 3.4.3.4 arrangement. The pattern lines applied to the projected triangular and square faces each have two-dimensional analogues, and each of these analogous two-dimensional patterns was used historically. A Mamluk example of the two-dimensional use of the design contained within just the triangular repetitive cells is found at the Ribat of Ahmad ibn Sulayman al-Rifa’i in Cairo (1291) [Fig. 300b two-point]; and a Mamluk example of the analogous two-dimensional square pattern was used in the triangular side panels of the minbar at the Princess Asal Bay mosque in Fayyum, Egypt (1497-99) [Fig. 382]. For the patterns in these two repetitive cells to work together they must have identical edge conditions, and in this respect, this domical design shares the characteristics of two-dimensional hybrid designs. Indeed, the two-dimensional forms of these two repetitive cells will comfortably tessellate the plane in either a 32.4.3.4 or 33.42 semi-regular arrangement [Fig. 89].

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Fig. 502

Other domical geometric designs based on the symmetry of polyhedra include two examples from the Alhambra. The Nasrid barrel vault of the Sala de la Barca (c. 1354-91) is capped at both ends with a quarter spherical rhombicuboctahedron. This polyhedra has 26 spherically projected faces: 8 that are triangles and 18 that are squares. The 24 identical vertices are a 3.43 arrangement of these faces. The pattern in the square faces places 12-pointed stars at the center of the repetitive face, 8-pointed stars at the four vertices, and 8-pointed stars at the midpoints of each repetitive edge. The triangular faces also have eight-pointed stars at the midpoints of the repetitive edges and the triangular vertices. The repetitive schema of the pattern in the barrel-vaulted portion of this ceiling is a simple square grid that transforms into the rhombicuboctahedron at each end of the barrel vault. The rings of eight square faces that characterize this polyhedra elegantly allow for this transformation. Unlike the earlier Ayyubid and Zangid wooden polyhedral domical constructions, the geometric design in this Nasrid wooden vault displays significant distortion. For example, the above-referenced placement of eight-pointed stars at the vertices of the repetitive cells works nicely among the square repeat of the barrel-vaulted portion of this ceiling, but is incompatible with the symmetry of the triangle and produces poorly resolved features in the rhombicuboctahedron semi-dome regions of the vault. In essence, the artists responsible for this example were conceptually inspired in their combined use of the barrel vault and rhombicuboctahedron, but underwhelming in their application of the geometric design. Ironically, had the same basic geometric pattern been applied to the polygonal faces with the 12-pointed stars placed upon the vertices rather than at the centers of each face, their 12-fold symmetry would have accommodated the vertices of both the barrel vault and polyhedral regions very acceptably. Indeed, this is exactly the approach to the polyhedral pattern in one of the two domes that grace the nearby Court of the Lions (c. 1354-91). This Nasrid example also employs spherical projections of square and triangular faces. This exceptional polyhedral dome is unusual in that it does not adhere to the geometry of either the Platonic or Archimedean solids. Rather, as discussed in Chap. 1, this is essentially an octacapped truncated octahedron with two varieties of triangle rather than a single type of non-equilateral triangle.Footnote 89 This polyhedra has 6 square faces, 40 equilateral triangular faces, and 8 triangular faces with a single 30° acute angle and two 75° obtuse angles. The analysis of Emil MakovickyFootnote 90 illustrates the unusual face and vertex configurations of this polyhedral dome, as well as the application of the pattern lines into these repetitive modules. This polyhedron has three varieties of vertex condition. The top half of this hemispherical dome follows the face configuration of the snub cube. Each vertex of this Archimedean solid has a 34.4 arrangement of triangles and square. In applying the pattern lines into this vertex configuration, the placement of three points of a star at each square faced vertex and two points at each triangular face produces 11-pointed stars [3 + (2 × 4) = 11]. This 11-fold symmetry at the polyhedral vertices is distinct from the 12-fold local symmetry that results from the same distribution of points within two-dimensional patterns that employ square and triangular repetitive cells [Fig. 23d]. The second variety of vertex in this unusual polyhedra has five projected equilateral triangular faces combined with the 30° acute angle of the isosceles triangle. This 30° angle provides for one point of a 12-pointed star on the flat plane, and the applied pattern lines in this arrangement thereby also produce an 11-pointed star [1 + (2 × 5) = 11]. The third vertex of this polyhedron combines one square, two equilateral triangles, and two of the 75° obtuse angles from the edge-to-edge isosceles triangles. These back-to-back triangular faces produce a rhombus with a 150° included angle, thereby providing for five points of a 12-pointed star in two dimensions. In this way, the third variety of vertex also has 11-pointed stars [3 + (2 × 2) + 5 = 11]. This spherical variant of the octacapped truncated octahedron has bilateral symmetry wherein the top and bottom halves of the sphere are mirrored upon the equatorial plane. In this way, the pattern lines at the base of the dome are nicely resolved along the equator. This polyhedron does not project to a spherical surface without a small amount of distortion among the equilateral triangular faces. Had the artist responsible for this remarkable dome chosen to use the snub cube as the repetitive schema, thereby continuing the thematic use of the 11-pointed stars at each vertex, the resolution of the pattern lines at the base of the dome would not have been as successful, and it is likely that the development of the octacapped truncated octahedral variant stemmed from the desire to resolve the pattern at the base of the hemispherical dome in a very acceptable fashion while maintaining the regular distribution of 11-pointed stars that the snub cube also provides.

Nasrid artists employed the truncated cube as the governing geometry for the spherical pommel of the Jineta sword of Muhammad XII, the last Muslim ruler in Spain. This Archimedean polyhedra is comprised of six octagonal faces and eight triangular faces in a 3.82 condition at each vertex. This is analogous to the two-dimensional 4.82 tessellation of squares and octagons, and the resulting spherical pattern is generated in the same fashion as the well-known star-and-cross design [Fig. 124b] whereby the octagonal faces produce eight-pointed stars. However, the replacement of the squares with triangles means that the eight-pointed stars are separated by trifold elements rather than fourfold crosses. Compared to the other Nasrid polyhedral geometric designs, this is considerably less complex, and its beauty is derived more from the opulence of the cloisonné enamel work than from its geometric ingenuity.

3.6 Conclusion

This survey of the methodological practices encompassed by the polygonal technique reveals the historical development from simplicity to ever-greater complexity. This trajectory built upon the early innovations and experimental approach of artists during the eighth to eleventh centuries to evolve into the highly formalized design conventions of the fully mature tradition during the twelfth to fourteenth centuries. Trained in these formalized methodological practices, Muslim artists expanded this tradition to include highly ingenious nonsystematic patterns with multiple regions of local symmetry, and systematic dual-level designs with varying degrees of self-similarity. Throughout this growth in complexity, nuanced design practices expanded the aesthetic repertoire available to artists. Some of these practices can be regarded as general rules and others as stylistic conventions. Yet even those practices that are constant features of the polygonal technique were occasionally modified or dispensed with, and the one criteria that applies above all others is whether a given design falls comfortably within the prevailing aesthetics of the time and place. The following brief encapsulation of the methodological practices inherent to this tradition provides fundament criteria to the nuanced understanding of this ornamental tradition, as well as valuable methodological praxis for contemporary artists and designers engaged with this geometric art form.

  • The polygonal technique is comprised of two methodological categories: systematic and nonsystematic. Both rely on underlying polygonal tessellations upon which pattern lines are distributed. Systematic designs employ a limited set of polygonal modules, each with prescribed pattern lines, which assemble into an unlimited number of combinations. By contrast, the proportions of underlying polygons that comprise nonsystematic designs are specific to each tessellation and do not recombine into other tessellations.

  • There are five historical design systems: the system of regular polygons, the fourfold system A, the fourfold system B, the fivefold system, and the sevenfold system. Different Muslim cultures had greater or lesser affinities with each of these design systems.

  • There are four primary historical pattern families common within this design tradition: acute, median, obtuse, and two-point. Each underlying tessellation is capable of producing patterns in each of these pattern families, although some generative tessellations are not suited to produce acceptable designs in each family.

  • Acute, median, and obtuse crossing pattern lines are applied to the midpoints of the underlying polygonal edges unless the pattern is improved by moving the crossing pattern lines to a nearby point along the polygonal edge. Generally, the placement of the crossing pattern lines within systematic designs maintains their location upon the midpoints of the underlying polygonal edges. In more complex nonsystematic patterns, the crossing pattern lines may be move off of the polygonal edge if this provides a more acceptable visual result.

  • The location of the pattern lines in the two-point family can vary, and, depending on the resulting aesthetic quality, 1/3 and 1/4 divisions of the polygonal edges are both common.

  • As pattern lines cross one another, they should ideally continue in a straight line beyond the intersection. Pattern lines that change direction at the point of intersection, thereby loosing their collinearity, generally appear awkward and are best avoided whenever possible.

  • In placing the crossing pattern lines on or near the midpoint of the underlying polygonal edge, the angular opening of the crossing pattern lines determines whether the design will be of the acute, median, or obtuse pattern family. With systematic patterns the precise angle is determined by the inherent geometry of the given system. With nonsystematic patterns the angular openings are ultimately determined through aesthetic judgment, and may vary slightly from location to location. Generally, the inherent angles of the fivefold system provide a useful comparative aesthetic when working with nonsystematic designs.

  • Other than designs with radial symmetry, historical Islamic geometric designs invariably employ translation symmetry, and will adhere to one of the 17 plane symmetry groups. A diverse range of repeat units were used historically, including; squares, regular hexagons, rectangles, rhombi, and non-regular hexagons. Equilateral triangles were frequently used as repetitive cells, but must be either mirrored to form a rhombus, or rotated to form a hexagon before providing translation symmetry.

  • All repetitive geometric patterns have a fundamental domain that contains all of the geometric information necessary to complete the design. The singular or combined application of rotation, reflection, and glide reflection to the fundamental domain fills the repeat unit, allowing for translation symmetry. In some cases, the fundamental domain repeats with translation symmetry alone.

  • Hybrid patterns can be constructed from using two or more repetitive cells in combination. This design methodology creates greater complexity within the completed pattern. Typically, each of these repetitive cells will tessellate the plane independently. A criteria of each repetitive cell within a given hybrid design is that all edges of equal length share the same pattern conditions and underlying generative tessellation conditions.

  • As a general rule, the n-fold symmetry of primary stars will correspond with the symmetry of its location within the repetitive structure. For example, orthogonal patterns will place stars with points that are multiples of four at the vertices of the square grid; isometric patterns will place stars with points that are multiples of six at the vertices of the triangular grid, and regular hexagonal grids will place stars that are multiples of three at each vertex. The same rule applies to the centers of each repetitive cell. The reflection symmetry of stars placed upon an axis of reflection, such as the edges and diagonals of a repeat unit, must align with the axis of reflection, and may be either even or odd numbered.

  • The proportions of rectangular repeat units are determined by the n-fold local symmetries placed at each vertex of the rectangular grid, and, where relevant, by the n-fold symmetries of secondary locations, such as the center and diagonals, within the repeat unit.

  • The proportions of rhombic and non-regular hexagonal repetitive grids are determined by the internal angles of each repeat unit corresponding with the n-fold symmetry of the primary stars placed at these locations.

  • Field patterns have no primary stars. This category of pattern is created from underlying generative tessellations that do not include primary polygons with a larger number of edges.

  • The flexibility of oscillating square and rotating kite designs allows for significant geometric manipulation, including the incorporation of unexpected n-fold local symmetries into an orthogonal structure. Such designs generally conform to the p4g plane symmetry group.

  • Framing rectangles almost always adhere to the geometry of the repetitive grid. In determining the frame for a given pattern, the edges should ideally correspond with lines of symmetry within the overall pattern whenever possible.

  • There are many forms of line treatment that can be applied to the basic plain line version of a given design. Each has its own aesthetic quality, and may be more or less appropriate to a given design, the aesthetic sensibilities of an artist, and different materials and techniques of fabrication. Regional styles can also be influenced by line treatment. Such treatments include tiling with two or more colors (as per a chess board), various thicknesses of widened lines with or without interweaving, and various forms of double-line applications.

  • The widening of pattern lines can be derived from key points within the polygonal sub-grid, or can be a purely arbitrary decision based upon visual preference. The thickness can also be determined from the geometry of design itself. For example, the pattern lines can be widened to their maximum extent, corresponding to the center of the smallest background element.

  • Patterns that are unsuccessful due to large discrepancies in the size of the background shapes can sometimes be corrected by the widening of the pattern lines with a single sided offset. This reduces the size of the larger elements while keeping the smaller elements the same size.

  • The primary stars in systematic designs are generally only of one variety: 8-pointed for both the fourfold system A and fourfold system B, 10-pointed for the fivefold system, and 14-pointed for the sevenfold system. Occasionally, stars with double the number of points will be incorporated into patterns created from these systems. Patterns created from the system of regular polygons will have 6-pointed and/or 12-pointed stars; and the primary stars of patterns created from the 4.82 tessellation of squares and octagons will invariably be 8-pointed.

  • Patterns can be modified through either additive or subtractive processes. These generally involve adding a secondary network of pattern lines that fit within the original design, but are independent of the underlying tessellation, or the removal of portions of the original set of pattern lines. In either case, these are arbitrary modifications, often subject to cultural predilections, that can substantially change the appearance of a given design.

  • The lines of the primary stars in each of the five historical systems can be modified in several fashions, leading to distinctive stylistic variation of a given design. This can include an infill whereby a pattern with primary stars is transformed into a field pattern. The same type of modification to the primary stars can also be applied to nonsystematic designs.

  • The primary stars of median patterns, two-point patterns, and occasionally acute patterns, can be modified such that the outer points are replaced with lines that extend into the primary polygons and form a new star rosette. This modification is especially associated with Mamluk aesthetics.

  • Another form of pattern modification involves replacing the straight lines of a design with curvilinear lines. This produces a floriated variation that can be comprised of circles, arcs, and s-curves.

  • In creating underlying tessellations with the modules from one of the design systems, interstice regions are sometimes produced. These will often produce satisfactory pattern characteristics in some, but not necessarily all of the four pattern families.

  • The system of regular polygons makes use of underlying generative tessellations comprised of regular polygons, including the triangle, square, hexagon, and dodecagon. Some patterns created from this system also include a hexagonal ditrigon that is derived from six overlapping squares. These underlying generative tessellations correspond with the regular grids, semi-regular grids, two-uniform grids, and three-uniform grids.

  • There is greater historical variability in the angular openings of the crossing pattern lines within the system of regular polygons. The acute angles within this system have 30° crossing pattern lines; the median have either 60° or 90°; and the obtuse have either 120° or 135°. Similarly, there are more types of two-point pattern line application within this system. The crossing pattern lines within the other four historical systems are generally limited to a single variety for each pattern family.

  • In some of the earlier designs created from the system of regular polygons, different polygonal cells within the overall underlying tessellation were treated as either active of passive. The active cells are used to generate the pattern lines, whereas the pattern lines extend into the passive cells from their active neighbors.

  • The 4.82 tessellation of squares and octagons is the generative basis for a very large number of historical designs. This tessellation is one of the semi-regular grids, and as such, this can qualify as part of the system of regular polygons. However, this is the only tessellation created from regular polygons that includes the octagon (it will not tessellate with the other regular polygons in any other manner). The octagon and square are also components of the fourfold system A. The fact that these two modules are shared by both systems, and that they have been used historically to create so many distinctive patterns, provides them with a stand-alone quality for separate consideration.

  • The shape and proportions of the secondary polygonal modules in the fourfold system A and fivefold system are easily derived from their primary polygon: the octagon and decagon respectively. Within the fourfold system B and the sevenfold system, the secondary polygonal modules are derived primarily from interstice regions that occur when tessellating with already established modules. Modules can also be created through the truncation of other modules, through overlapping other modules, and through the union of other modules.

  • The underlying primary polygons in both systematic and nonsystematic tessellations occasionally overlap with one another, creating a larger conjoined polygon. This creates dual star forms that are often very satisfactory in one or another pattern family.

  • In each of the polygonal systems, the dual of some underlying tessellations are also comprised of polygonal modules from the same system. In such incidences, each will create the same geometric pattern. In this way, the same geometric design can often be created from more than a single underlying tessellation.

  • Some configurations of polygons work very well with one or two of the pattern families, but very poorly with the others. For example, the arrangement of six pentagons surrounding a thin rhombus works well with the obtuse and two-point families but not with the acute and median families. In such cases, the underlying pentagons can be changed into trapezoids to produce successful designs in these latter two families. This same principle applies to nonsystematic design methodology.

  • The modules of the fivefold system can be used at two different scales to create a single level pattern with variable pattern density. There are two edge lengths in the modules that comprise the fivefold system. The ratio of these edge lengths is phi: the golden section (1:1.6180…). The diminishing scales of the two historical examples that employ this device are based upon this proportion. Historical examples of this type of systematic design are very rare, but this scaling feature offers tremendous scope to contemporary artists.

  • Nonsystematic design methodology involves three phases: (1) the creation of the radii matrix; (2) the creation of the underlying tessellation; and (3) the creation of the geometric design. Each succeeding phase is directly dependant on its predecessor. Radii matrices are fundamental to the nonsystematic design process and examples of their historical use are found within the Topkapi Scroll.

  • In each of the four pattern families, the fivefold system provides the aesthetic criterion for achieving success in producing nonsystematic designs. Applying methodological conventions established in the fivefold system to nonsystematic pattern making allows for greater design flexibility and diversity. For example, modifications to underlying tessellations that are standard to the fivefold system will work analogously in similar nonsystematic situations. In particular, the ring of pentagons that typically surround the decagons of the fivefold system is an immensely successful formative device that can also be used to great effect with nonsystematic patterns.

  • Nonsystematic patterns can have a single variety of primary star or multiple star forms. Among the most interesting nonsystematic designs are those that employ several star forms that are in numeric sequence, such as 9-, 10-, 11-, and 12-pointed stars. Also of particular interest are patterns the employ the principle of adjacent numbers. Such patterns work on the premise that if an individual star with n-fold rotation symmetry works particularly well for constructing patterns, then patterns that have both (n + 1)-fold symmetry and (n − 1)-fold symmetry can also create acceptable patterns. In this way, since eight-pointed stars make very good designs, patterns with seven- and nine-pointed stars should also work well together. Similarly, 10- provides for 9- and 11-pointed stars; and 12- provides for 11- and 13-pointed stars.

  • Nonsystematic patterns are particularly flexible in the varieties of repetitive cells that they encompass. Of course the square, equilateral triangle, and regular hexagon are especially common, but rectangular, rhombic, and non-regular hexagons are also represented. A limited number of patterns with radial symmetry are also known to the historical record. The inherent flexibility of this design tradition extends into the realm of repetitive grids, and there are many new approaches that open the door to original designs.

  • Each of the five historical design systems lend themselves to the creation of dual-level designs with self-similar characteristics. This methodology involves scaling down the polygonal modules (with their applied pattern lines) at a ratio that allows these smaller modules to seamlessly populate the primary pattern lines and background regions.

  • There are four historical varieties of dual-level design. Type A designs have a single line for the primary pattern with full coverage of the secondary pattern. Type B designs have widened lines for the primary pattern that are filled with the secondary pattern. Type C designs also employ widened lines of the primary pattern, but the secondary pattern fills both the widened lines (as per type B) as well as the background regions, thereby providing full surface coverage (as per type A). Type D dual-level designs are native to the Maghreb and differentiate the primary and secondary patterns through color.

  • Each variety of dual-level design involves the placement of scaled-down polygonal modules with associated pattern lines from one or another of the five historical design systems onto key locations of the primary pattern or its underlying tessellation. The scaling ratio is determined by the secondary polygonal modules fitting edge-to-edge into the primary pattern matrix, and is always a factor of the proportional relationships inherent within the given design system: for example, phi for the fivefold system.

  • The self-similarity of historical dual-level designs is rarely scale invariant, but isolated regions within a given design will occasionally have scale invariance. More often, the self-similarity of recursive multilevel designs is a product of their use of the same pattern family, with identical design characteristics at each successive level.

  • The dual-level design methodology can, in theory, be applied recursively to multiple levels ad infinitum.

  • Although no examples of true quasicrystalline designs are known to the historical record, the recursive dual-level design methodology can be used to create designs that meet the criteria of aperiodicity and quasicrystallinity. Such designs can be constructed so that they adhere to the Penrose matching rules and incorporate inflation and deflation.

  • There are two historical conventions for applying geometric designs to the surfaces of domes. The most common employs gore segments as the radial repeat unit. The second method is rarely encountered, and projects the polyhedral symmetries of the Platonic and Archimedean solids onto the domical surfaces. The Mughals in India practiced a third technique wherein 2/10 of a two-dimensional tenfold radial pattern were removed from the design and the remaining 8/10 were closed to form a cone. This was then applied, with distortion, to the surface of the dome.

  • If there is one overarching principle that is responsible for the longevity and success of this design tradition and is, indeed, fundamental to the revival of this artistic discipline it is innovation. The codified practices as outlined above are a means of working within established aesthetic constraints, but these “rules” are always open to bending and even breaking if the results are beautiful and expand the aesthetic horizons into uncharted territories. The need for innovation is no less relevant today than it was throughout the history of Islamic geometric pattern.

This conclusion is actually just the beginning of an endless quest to better understand and appreciate methodological approaches to creating beauty with geometry. May all who endeavor along this path find creative inspiration from the geometry that permeates our world and universe, from the masterpieces of geometric art of virtually every culture, and from the deep rooted affinity for geometry that is inherent within the human mind and heart.