Keywords

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.

Fig. 1.
figure 1

The side portal of the cathedral of Prato. The two topmost figures have octagonal shape: the one on the right is based on a regular octagon, while the one on the left seems to allude to a very particular construction that inspires thus paper and the now called Carboncettus octagons.

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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.,

$$\begin{aligned} F_0=0,\quad F_1=1,\quad F_2=1,\quad F_3=2,\quad F_4=3,\quad F_5=5,\quad \text {etc.} \end{aligned}$$

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 \(C_1\).

Fig. 2.
figure 2

The construction of the Carboncettus octagon. In the picture, in particular, it is shown in red the octagon \(C_1\). (Color figure online)

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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

$$\begin{aligned} {OA},\ {OB},\ {OC},\ {OD},\ {OE},\ {OF},\ {OG},\ {OL} \end{aligned}$$
(1)

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

$$\begin{aligned} \begin{array}{lll} |{AB}|= |{OA}|=2, &{}\qquad \qquad \quad &{} |{OH}|=|{KC}|=\sqrt{3}, \\ |{BC}|=\sqrt{6} -\sqrt{2}, &{}\qquad \qquad \quad &{} |{ON}|=\frac{\sqrt{6} +\sqrt{2}}{2}, \end{array}\end{aligned}$$
(2)

which, for example, are all incommensurable in pairs. Instead, for the widths of the angles we trivially have

$$\begin{aligned} \begin{array}{c} \measuredangle AOB = \measuredangle OBA = \pi /3=60^{\circ }, \qquad \measuredangle BOC = \measuredangle HOB = \pi /6=30^{\circ }, \\ \measuredangle OBC = 5\pi /12=75^{\circ }. \end{array}\end{aligned}$$
(3)

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

$$\begin{aligned} \mathrm{Per}(C_1)= 8+4\sqrt{6}-4\sqrt{2}, \qquad \mathrm{Area}(C_1)=4+4\sqrt{3}. \end{aligned}$$

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.

Table 1. Some metric data relative to the first three elements of the Carboncettus sequence, after \(C_1\). The letters are displayed in the construction as in Fig. 2.
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Fig. 3.
figure 3

The second element of the Carboncettus sequence, the octagon \(C_2\), is drawn in black. A regular octagon inscribed in the same circumference \(\varGamma _3\), is also represented in red. (Color figure online)

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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

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{F_n}{F_n}=1 \qquad \text { and } \qquad \lim _{n\rightarrow \infty } \frac{F_{n+2}}{F_n}, \end{aligned}$$
(4)

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

$$\begin{aligned} \phi := (1+\sqrt{5})/{2} \approx 1.618033987, \end{aligned}$$
(5)

hence, the second limit in (4) is simple to compute as followsFootnote 3

$$\begin{aligned} \lim _{n\rightarrow \infty } \frac{F_{n+2}}{F_n}\ =\ \lim _{n\rightarrow \infty } \frac{F_{n+2}}{F_{n+1}}\cdot \frac{F_{n+1}}{F_n}\ =\ \phi ^2\ \approx \ 2.618033987, \end{aligned}$$
(6)

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]).