Abstract
Considering the classic Fibonacci sequence, we present in this paper a geometric sequence attached to it, where the word “geometric” must be understood in a literal sense: for every Fibonacci number \(F_n\) we will in fact construct an octagon \(C_n\) that we will call the n-th Carboncettus octagon, and in this way we obtain a new sequence \(\big \{C_n \big \}_{n}\) consisting not of numbers but of geometric objects. The idea of this sequence draws inspiration from far away, and in particular from a portal visible today in the Cathedral of Prato, supposed work of Carboncettus marmorarius, and even dating back to the century before that of the writing of the Liber Abaci by Leonardo Pisano called Fibonacci (AD 1202). It is also very important to note that, if other future evidences will be found in support to the historical effectiveness of a Carboncettus-like construction, this would mean that Fibonacci numbers were known and used well before 1202. After the presentation of the sequence \(\big \{C_n\big \}_{n}\), we will give some numerical examples about the metric characteristics of the first few Carboncettus octagons, and we will also begin to discuss some general and peculiar properties of the new sequence.
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1 Introduction
The names here proposed of “n-th Carboncettus octagon” and “Carboncettus sequence/family of octagons”, or better, the inspiration for these names, comes from far away, sinking its roots in the early centuries of the late Middle Ages. They are in fact connected to the cathedral of Prato, a jewel of Italian Romanesque architecture, which underwent a profound restructuring in the 11th century, followed by many others afterwards. The side portal shown in Fig. 1 (which we will later call simply the portal) at the time of its construction seems to have been the main portal of the cathedral. The marble inlays on its sides and the figures represented have aroused many discussions among scholars for many years and in particular have always aroused the attention and interest of G. Pirillo, an interest that he recently transmitted also to the other authors. Pirillo studied the figures of the portal for a long time and traced a fascinating symbolism, typical of medieval culture (see for example [11]). According to these studies, the right part of the portal, for instance, through a series of very regular and symmetrical figures, would recall the divine perfection, while the left part, through figures that approximate the regular ones but are not themselves regular, the imperfection and the limits of human nature. The very interesting fact is that the artist/architect who created the work (which is thought to be a certain Carboncettus Marmoriarius, very active at that time and in those places, [11]) seems to have been in part used the mathematical language to express these concepts and ideas, and this thing, if confirmed, would assume enormous importance, because before the 12th century we (and many experts of the field) have no knowledge of similar examples. The construction of the Carboncettus octagon (or better, of the Carboncettus octagons, since they are infinitely many) originates from Fibonacci numbers and yields a sequence not of numbers but of geometrical figures: we will explain the details starting from Sect. 2.
From the historical point of view we cannot avoid to note an interesting, particular coincidence: probably, the most known and most important octagonal monument existing in Calabria dates back to the same period as the construction of the portal of the Duomo of Prato, and it is the octagonal tower of the Norman-Swabian Castle in Cosenza. But it is important to specify, for the benefit of the reader, that, in Cosenza, on the site of the actual Norman-Swabian Castle, a fortification had existed from immemorial time, which underwent considerable changes over the years: first a Bruttuan fortress, then Roman, Norman and Swabian, when it had the most important restructuring due to Frederick II of Swabia. In particular, it is Frederick who wanted the octagonal tower visible today, his preferred geometric shape: remember, for example, the octagonal plan of the famous Castel del Monte near Bari, in Apulia.
With regard to Fibonacci numbers, we would like to point out to the reader for completeness of information, a recent thesis by G. Pirillo often and many times discussed within this group of authors. In [10, 12,13,14] Pirillo presented the audacious thesis that the first mathematicians who discovered Fibonacci numbers were some members of the Pythagorean School, well documented and active in Crotone in the 6th, 5th and 4th centuries B.C., hence about 1,700 years before that Leonardo Pisano, known as “Fibonacci”, wrote his famous Liber Abaci in 1202. Such a thesis is mainly supported by computational evidences arising from pentagon and pentagram about the well-known Pythagorean discovery of the existence of incommensurable numbers. The interested reader can find further information and references on the Pythagorean School, incommensurable lengths, Fibonacci numbers and some recent developments in [6, 8, 10, 14,15,16,17].
Similarly to the above thesis note that, since the portal in Prato is dating back to the 12th century, if other future evidences will support the employ of Fibonacci numbers in its geometries, this would mean that they were known before 1202 as well, even if only a few decades.
A final remark on notations: we denote by \(\mathbb {N}\) the set of positive integers and by \(\mathbb {N}_0\) the set \(\mathbb {N}\cup \{0\}\). A sequence of numbers or other mathematical objects is denoted by \(\big \{a_n\big \}_{n\in \mathbb {N}}\), \(\big \{a_n\big \}_{n}\), or simply \(\{a_n\}\). If, moreover, A, B, C are three points of the plane, AB denotes the line segment with endpoints A and B, \(\left| {AB}\right| \) its length, and \(\measuredangle {ABC}\) the measure of the angle with vertex in B.
2 The Carboncettus Family of Octagons
If r is any positive real number, we denote by \(\varGamma _r\) the circumference of radius r centered in the origin. As usual, let \(F_n\) be the n-th Fibonacci number for all \(n\in \mathbb {N}_0\), i.e.,
If \(n\in \mathbb {N}\) we consider a couple of concentric circumferences having radii of length \(F_n\) and \(F_{n+2}\), respectively. If \(n=1\) they are represented in green in Fig. 2, were the radius of the inner circumference is 1 and that of the outer one is 2, i.e. \(F_{3}\). Then we draw two couples of parallel tangents, orthogonal between them, to the inner circumference and we consider the eight intersection points A, B, C, D, E, F, G, L with the outer circumference \(\varGamma _{F_{n+2}}\), as in Fig. 2. The octagon obtained by drawing the polygonal through the points A, B, C, D, E, F, G, L, A, in red in Fig. 2, is called the n-th Carboncettus octagon and is denoted by \(C_n\). Therefore, the red octagon in Fig. 2, is the first Carboncettus octagon \(C_1\).
From a geometrical point of view, the Carboncettus octagon \(C_n\) is more than a cyclic polygon; it is in fact an isogonal octagon for all \(n\in \mathbb {N}\), that is, an equiangular octagon with two alternating edge lengths.Footnote 1 More recently it is also used to say a vertex-transitive octagon: all the vertices are equivalent under the symmetry group of the figure and, in the case of \(C_n\), for every couple of vertices, the symmetry which send the first in the second is unique. The symmetry group of \(C_n\) is in fact isomorphic to the one of the square, the dihedral group \(D_4\).Footnote 2
An interesting property of the Carboncettus sequence \(\big \{C_n\big \}_{n\in \mathbb {N}}\) is the fact that, with the exception of the first three elements \(C_1,\ C_2,\ C_3\) (or, at most, also \(C_4\)), all the subsequent ones are completely indistinguishable from a regular octagon (see, for example, Fig. 3 representing \(C_2\): it is yet relatively close to a regular octagon). Due to the lack of space, we will deepen these and other important aspects mentioned in the following, in a subsequent paper in preparation.
3 The First Four Octagons of the Carboncettus Sequence: Geometric Properties and Metric Data
In this section we will give some numerical examples looking closely at the first elements of the sequence \(\big \{C_n\big \}_{n\in \mathbb {N}}\).
Example 1
(The octagon \(C_1\)). The first Carboncettus octagon \(C_1\) is built starting from the circumferences \(\varGamma _1\) and \(\varGamma _2\), as said in Sect. 2. In this case we obtain a very particular isogonal octagon: drawing the eight radii
of the circumference \(\varGamma _2\) as in Fig. 2, the resulting shape has commensurable angle measures, in fact all them are integer multiples of \(\pi /12=15^{\circ }\). Not only; in this way \(C_1\) results formed by 4 equilateral triangles (congruent to ABO, see Fig. 2) and 4 isosceles triangles (congruent to BCO). The lengths of their sides and heights are
which, for example, are all incommensurable in pairs. Instead, for the widths of the angles we trivially have
Discussing the commensurability of the angles for all the sequence \(\big \{C_n\big \}_{n}\) is interesting, but we are forced to postpone this elsewhere. The same, as well, considering the commensurability, along all the sequence, of some of the side lengths made explicit in (2). Note lastly that perimeter and area are
The second Carboncettus octagon \(C_2\) originates from the circumferences \(\varGamma _1\) and \(\varGamma _3\), with radii \(F_2=1\) and \(F_4=3\), respectively, and the result is the black octagon in Fig. 3, compared with a red regular one inscribed in the circumference \(\varGamma _3\) itself. Using the letters disposition of Fig. 2, the lengths of the correspondent sides and heights considered in (2), the angle widths, perimeter and area, are those listed in the second column of Table 1.
4 The “limit octagon” and Future Researches
Many aspects of the new sequence \(\big \{C_n\big \}_{n}\) are interesting to investigate. For example, scaling the octagon \(C_n\) by a factor equal to the n-th Fibonacci number \(F_n\), the sequence will converge to a limit octagon \(C^N_{\infty }\) (where the top N stands for “normalized”) that can be drawn through the “Carboncettus construction” described at the beginning of Sect. 2, by starting from the circumferences with radii given by the following limit ratios
respectively. It is well known that the limit of the ratio of two consecutive Fibonacci numbers \({F_{n+1}}/{F_n}\) converges to the golden ratio
hence, the second limit in (4) is simple to compute as followsFootnote 3
and we conclude that \(C^N_{\infty }\) can be constructed using the circumferences \(\varGamma _1\) and \(\varGamma _{\phi ^2}\).
Another approach to directly study the “limit octagon” \(C_{\infty }\) instead of the “limit normalized octagon” \(C^N_{\infty }\), could come by using the computational system introduced for example in [18,19,20] and applied as well to limit curves, limit polytopes, fractals and similar geometric shapes in [1, 3,4,5, 20] (or even to Fibonacci numbers in [9]).
Notes
- 1.
- 2.
Note, for didactic purposes, how the multiplication table of \(D_4\) emerges much more clearly to the mind of a student thinking to \(C_1\) than thinking to a square.
- 3.
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This work is partially supported by the research projects “IoT&B, Internet of Things and Blockchain”, CUP J48C17000230006, POR Calabria FESR-FSE 2014–2020.
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Caldarola, F., d’Atri, G., Maiolo, M., Pirillo, G. (2020). The Sequence of Carboncettus Octagons. In: Sergeyev, Y., Kvasov, D. (eds) Numerical Computations: Theory and Algorithms. NUMTA 2019. Lecture Notes in Computer Science(), vol 11973. Springer, Cham. https://doi.org/10.1007/978-3-030-39081-5_32
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