New algebraic and geometric constructs arising from Fibonacci numbers

Abstract

Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family $\{C_n:n\in \mathbb {N}\}$ of octagons that come very close to the regular octagon. Such octagons, in some previous articles, have been given the name of Carboncettus octagons for historical reasons. Going further, in this paper we want to introduce and investigate some algebraic constructs that arise from the family $\{C_n:n\in \mathbb {N}\}$ and therefore from Fibonacci numbers: From each Carboncettus octagon $C_n$, it is possible to obtain an infinite (right) word $W_n$ on the binary alphabet $\{0,1\}$, which we will call the nth Carboncettus word. The main theorem shows that all the Carboncettus words thus defined are Sturmian words except in the case $n=5$. The fifth Carboncettus word $W_5$ is in fact the only word of the family to be purely periodic: It has period 17 and periodic factor 000 100 100 010 010 01. Finally, we also define a further word $W_{\infty }$ named the Carboncettus limit word and, as second main result, we prove that the limit of the sequence of Carboncettus words is $W_{\infty }$ itself.

1 Introduction

In this paper, we associate with each Fibonacci number $F_n,\ n\ge 1$, a geometric construct $C_n$ and then an algebraic object $W_n$, obtaining simultaneously three sequences $\left\{ F_n\right\} _n$, $\left\{ C_n\right\} _n$ and $\left\{ W_n\right\} _n$, where the first one is the well-known sequence of integers, but the last two are new sequences not of numbers but of geometrical and algebraic objects, respectively. In particular, $\left\{ C_n\right\} _n$ is a sequence of octagons very close to a regular one and $\left\{ W_n\right\} _n$ is a sequence of infinite right words on the binary alphabet $\{0,1\}$. The reasons that led us to give them the name of Carboncettus octagons and words come, as we will see, from far away.

Prato is a Tuscan city located 17 km northwest of Florence. With its 200,000 inhabitants, Prato is the third largest city in central Italy, after Rome and Florence. The Cathedral of Prato, dedicated to the first Christian martyr Saint Stephen, is a jewel of Romanesque architecture of international appeal: It has ancient roots dating back to at least the sixth century AD and has undergone numerous renovations and modifications over the years, including, very important, those around the eleventh century. The current lateral portal of the Cathedral (Fig. 1) seems to have been the main portal at that time, and, in recent years, it has attracted the attention of various scholars, including several mathematicians, in an attempt to explain some figures inlaid in marble on the sides of the portal. It seems that on the right jamb of the portal, the regularity, symmetry and the subject of the figures recall the divine perfection and completeness, while on the left jamb, some almost regular figures seem to symbolize the limits of human nature, which is made, yes, in the likeness of God, but which is not divine, nor perfect as the divine one (for a deeper study into the rich symbolisms behind the representations on the sides of the portal, the reader can see Pirillo (2017a) and the references therein).

Fig. 1

The side portal of the Cathedral of Prato, dating back to the twelfth century AD. The comparison between the two tarsias located at the top of the jambs inspired the (hypothesis on the) construction of the Carboncettus octagon

G. Pirillo was fascinated by this portal and its typically medieval symbolism from the first time he saw it in the 70s. Above all, the two inlays with octagonal (or apparently octagonal) symmetry placed at the top of each jamb attracted him very much. The one on the right seems to be based on a perfect regular octagon, while the one on the left seems to allude to a different geometric construction of an octagon that uses two concentric circumferences. Pirillo recently noted that if the radii of the circumferences are equal to two Fibonacci numbers with indices of the same parity and consecutive, then, by means of a simple construction that uses two pairs of parallel tangents to the internal circumference, and perpendicular to each other, one obtains an octagon that is indistinguishable from a regular one, but which is not itself perfectly regular.¹ In few words, this can be viewed as the discovery of a beautiful approximate method to construct a (not perfectly precise) regular octagon.

It is also very important to note the following fact: The Liber Abaci by Leonardo Pisano called Fibonacci, where the $\left\{ F_n\right\} _n$ series bearing his name appears for the first time, dates back to 1202 AD, while the portal of the Cathedral of Prato, for as seen today, dates back to the previous century and seems to be the work of Carboncettus Marmoriarius, very active in the twelfth century in those places. If the use of Fibonacci numbers in the architecture of the portal in Prato were to be confirmed by other studies, this would be a fact of enormous importance for two main reasons:

• This would mean that the Fibonacci numbers were known at least a few decades before 1202;²

• The use of mathematical languages or tools in architecture to express concepts and ideas related to medieval (not only religious) symbolism seems not to have precedents before the twelfth century.

In the last two years, Pirillo has discussed many times the content of Pirillo (2017c) with the other authors of this work, and this has led to the article Caldarola et al. (2020c) on “The sequence of Carboncettus octagons.” In this paper instead, the main novelty is represented by the definition of an algebraic object for each octagon $C_n$ and, hence, for each Fibonacci number $F_n$, it is the right infinite word $W_n$ on the binary alphabet $\{0,1\}$ that will be precisely defined in Sect. 3 as a lower cutting sequence related to the extension of the height of a triangle which constitutes the octagon $C_n$ and will be named the nth Carboncettus word. Such an idea to introduce these algebraic objects follows an intuition of the first author about a peculiarity of the word $W_5$ as expressed in Theorem 1: All the words $W_n$ result to be Sturmian words except just in the single case $W_5$ where we obtain a periodic word with period 17 and periodic factor 000 100 100 010 010 01.

After the present introduction, this paper is organized into three central sections: the first two of which take up their titles from and deal with the mentioned geometric and algebraic aspects, respectively. In Sect. 2, we will introduce the geometric constructions, arising from Fibonacci numbers, that lead us to the family of Carboncettus octagons. In Sect. 3 instead, we will define, give some examples of and investigate the algebraic constructs obtained through the $\left\{ C_n\right\} _n$ family from the Fibonacci series as well: They are the Carboncettus words $W_n$, and Theorem 1, mentioned above, states our first main result on them. Finally, in Sect. 4 we will introduce the Carboncettus limit word $W_{\infty }$ and, as second main result of the paper, we will prove in Theorem 2 that the limit of the sequence of Carboncettus words is the Carboncettus limit word.

We inform the reader that we have tried to make this text self-contained and therefore suitable for reading even by those who are not specialists in the field. Only in the proof of Theorem 1, in fact, we do use three results that are not proven here.

As for notations, we use $\mathbb {N}$ for the set of positive integers and $\mathbb {N}_0$ for $\mathbb {N}\cup \{0\}$. A sequence is denoted by $\left\{ a_n\right\} _{n\in \mathbb {N}}$, $\left\{ a_n\right\} _{n}$, or sometimes simply $\{a_n\}$. If A, B, C are three points on the plane, we let AB denote the line segment with endpoints A, B, by |AB| its length and by $\measuredangle {ABC}$ the measure (in radians or degrees) of the angle with vertex in B.³

Fig. 2

The construction of the first Carboncettus octagon $C_1$

Fig. 3

The construction of the nth Carboncettus octagon $C_n$ for a general $n\ge 1$. In the picture, to avoid small and hard to read nested subscripts, we write $\varGamma '$ and $\varGamma ''$ instead of $\varGamma _{F_n}$ and $\varGamma _{F_{n+2}}$, respectively

2 New geometric constructions from Fibonacci numbers: the Carboncettus family of octagons

We recall that Fibonacci numbers are defined by $F_0:=0,\ F_1:=1$, and

$$\begin{aligned} F_{n}:= F_{n-2} +F_{n-1} \end{aligned}$$

(1)

for all $n\in \mathbb {N},\ n\ge 2$; in this way, we obtain the well-known sequence

$$\begin{aligned}&0,\,\ 1,\,\ 1,\,\ 2,\,\ 3,\,\ 5,\,\ 8,\,\ 13,\,\ 21,\,\ 34,\,\ 55,\,\ 89, \\&\quad 144,\,\ 233,\,\ 377,\,\ 610,\,\ 987,\,\ 1597,\,\ 2584,\,\ \text {etc.} \end{aligned}$$

For every real number $r>0$, we let $\varGamma _r$ denote the circumference in the plane $\mathbb {R}^2$, centered at the origin and with radius r. For every $n\in \mathbb {N}$, we consider the couple of Fibonacci numbers $(F_n,F_{n+2})$ and the circumferences $\varGamma _{F_n},\, \varGamma _{F_{n+2}}$, with radii $F_n$ and $F_{n+2}$, respectively. In the following, we will often refer to $\varGamma _{F_n}$ as the internal circumference and to $\varGamma _{F_{n+2}}$ as the external one. When $n=1$, $\varGamma _{F_1}=\varGamma _1$ and $\varGamma _{F_3}=\varGamma _2$ appear in green in Fig. 2, instead Fig. 3 depicts the case for general $n\ge 1$. We then draw a pair of horizontal tangents to the internal circumference $\varGamma _{F_n}$ through the two intersection points with the y-axis and a pair of vertical tangents through the two intersection points with the x-axis. These four straight lines, pairwise parallel or orthogonal, intersect the external circumference $\varGamma _{F_{n+2}}$ at eight points that we denote, starting from the first quadrant and proceeding counterclockwise, by $B_n,\, D_n,\, E_n,\, I_n,\, J_n,\, P_n,\, Q_n,\, A_n$ (Fig. 3). The nth Carboncettus octagon, denoted by $C_n$ ($n\ge 1$), is the one obtained by drawing the polygonal through the points

$$\begin{aligned} A_n,\ B_n,\ D_n,\ E_n,\ I_n,\ J_n,\ P_n,\ Q_n,\ A_n, \end{aligned}$$

and it is represented in red in Fig. 3.

Remark 1

Every Carboncettus octagon $C_n$ is obviously a cyclic polygon, that is, a polygon whose vertices all lie on the same circumference. But not only, $C_n$ is, more precisely, an isogonal octagon for all $n\ge 1$. An isogonal or vertex-transitive polygon is a polygon whose symmetry group acts transitively on the set of all its vertices. Therefore, an isogonal polygon is equiangular, cyclic and has at most two different alternating side lengths.

A very interesting property of the Carboncettus sequence $\left\{ C_n\right\} _{n\in \mathbb {N}}$ is that all its elements, with the exception of the first three or at most the first four, are completely indistinguishable from a regular octagon. See, for instance, Fig. 4 which compares the octagon $C_2$ with a regular one inscribed in the same circumference $\varGamma _{F_4}=\varGamma _3$: They are yet not too far one from the other.

Example 1

(The octagon $C_1$) The first Carboncettus octagon is obtained starting from the circumferences $\varGamma _{F_1}=\varGamma _1$ and $\varGamma _{F_3}=\varGamma _2$ (Fig. 2), and the resulting isogonal octagon $C_1$ has very particular characteristics. For instance, all the angles appearing in Fig. 2 are commensurable, because their measures are integer multiples of

$$\begin{aligned} \measuredangle B_1 ON_1=\frac{\pi }{12}=15^{\circ }. \end{aligned}$$

Moreover, $C_1$ results composed of four equilateral triangles congruent to $A_1 OB_1$ and four isosceles triangles congruent to $B_1 OD_1$. The lengths of their sides and heights are

$$\begin{aligned} \begin{array}{ll} |{A_1B_1}|= |{OA_1}|=2, &{} |{OH_1}|=|{K_1D_1}|=\sqrt{3}, \\ |{B_1D_1}|=\sqrt{6} -\sqrt{2}, &{} \displaystyle |{ON_1}|=\frac{\sqrt{6} +\sqrt{2}}{2}, \end{array} \end{aligned}$$

(2)

and the widths of the involved angles trivially are

$$\begin{aligned} \displaystyle \measuredangle A_1 OB_1= & {} \measuredangle OB_1A_1 = \frac{\pi }{3}=60^{\circ }, \nonumber \\ \displaystyle \measuredangle B_1 OD_1= & {} \measuredangle H_1 OB_1 = \frac{\pi }{6}=30^{\circ }, \\ \displaystyle \measuredangle D_1B_1 O= & {} 5\,\frac{\pi }{12}=75^{\circ }.\nonumber \end{aligned}$$

(3)

Note, for example, that, unlike the widths of the angles, all the lengths expressed in (2) are incommensurable in pairs. Lastly, perimeter and area of $C_1$ are as follows:

$$\begin{aligned}&\displaystyle {\mathrm{Perim}}(C_1)= 8+4\sqrt{6}-4\sqrt{2}, \\&\mathrm{Area}(C_1)=4+4\sqrt{3}.\nonumber \end{aligned}$$

(4)

Example 2

(The octagons $C_2,\ C_3$ and $C_4$) The construction of the second Carboncettus octagon is based on the circumferences $\varGamma _1$ and $\varGamma _3$, whose radii are $F_2=1$ and $F_4=3$, respectively. The obtained octagon $C_2$ is drawn in black in Fig. 4, where it is compared with a red regular octagon inscribed in the same circumference $\varGamma _3$. Some metric data concerning the octagon $C_2$ are listed in the second column of Table 1. Then, the third and the fourth column of the table do the same for the Carboncettus octagons $C_3$ and $C_4$, respectively.

Fig. 4

The octagon $C_2$, the second element of the Carboncettus sequence, is drawn in black starting from the circumferences $\varGamma _{F_2}=\varGamma _1$ and $\varGamma _{F_4}=\varGamma _3$. A regular octagon inscribed in the same circumference $\varGamma _3$ is instead represented in red

Much can be said, and there is much to study and investigate on the Carboncettus family of octagons $\{C_n:n\in \mathbb {N}\}$, but we will do it elsewhere because, as anticipated in the Introduction, we must now move on to the next section to introduce new algebraic constructs.

3 New algebraic constructs from Fibonacci numbers: the Carboncettus words $W_n$

The main reference for the general setting of this section is Lothaire (2002), but the reader can also see Lothaire (1983) and Berstel and Perrin (2007) for an essay on the recent origin of the field called combinatorics on words.

First of all, we need to recall a generalization of (1): A sequence $\left\{ G_n\right\} _{n\in \mathbb {N}_0}$ of integers (but also real or complex numbers) such that

$$\begin{aligned} G_{n}:= G_{n-2} +G_{n-1} \quad \text {for all } n\ge 2 \end{aligned}$$

is said a generalized Fibonacci sequence, and the couple $(G_0,G_1)$ is called the seed of the sequence. For example, the generalized Fibonacci sequence with seed (2, 1) is called Lucas sequence and denoted by $\left\{ L_n\right\} _{n\in \mathbb {N}_0}$. Hence, we have

$$\begin{aligned}&L_0=2,\quad L_1=1,\quad L_2=3,\quad L_4=4, \\&L_5=7,\quad L_6=11,\quad L_7=18,\quad \text {etc.} \end{aligned}$$

Table 1 Some metric data relative to the three Carboncettus octagons $C_2,\; C_3,\; C_4$, in the second, third and fourth column, respectively. Recall that the letters with the subscript n are displayed in the general construction of $C_n$ shown in Fig. 3

Now, we begin to talk about words, i.e., finite or infinite sequences of symbols from a finite alphabet A, starting with some definitions that we need for our purposes.

For any real numbers $\beta ,\; \rho$ with $\beta >0$, consider the line

$$\begin{aligned} L:\ \ y=\beta x+\rho \end{aligned}$$

(5)

in the plane $\mathbb {R}^2$, the grid $G:=\{(x,y)\in \mathbb {R}^2: x\text { or } y\in \mathbb {Z}, \text { and } x\ge 0\}$ in the right half plane $H:=\{(x,y)\in \mathbb {R}^2: x\ge 0\}$, and the sequence of intersection points of $L\cap G$

$$\begin{aligned} T_0= & {} (x_0,y_0)=(0,\rho ),\quad \ T_1=(x_1,y_1), \\ T_2= & {} (x_2,y_2),\quad \ T_3=(x_3,y_3),\quad \ \text {etc.}, \end{aligned}$$

where $0=x_0<x_1<x_2< \ldots$ and $L\cap G=\{T_n:n\in \mathbb {N}_0\}$.

Definition 1

• (i) From the sequence $\left\{ T_n\right\} _{n\in \mathbb {N}_0}$, we define an infinite word $K_{\beta ,\rho }$ by writing 0 for $T_0$ and continuing in succession for all $T_n,\ n\ge 1$, by writing

  • 0 if $x_n\in \mathbb {Z}$ and $y_n\notin \mathbb {Z}$;

  • 1 if $y_n\in \mathbb {Z}$ and $x_n\notin \mathbb {Z}$;

  • 10 if $x_n\in \mathbb {Z}$ and $y_n\in \mathbb {Z}$ too.

  $K_{\beta ,\rho }$ is called the lower cutting sequence attached to the line L.

• (ii) Similarly, we define an infinite word $K'_{\beta ,\rho }$ called the upper cutting sequence attached to the line L, by writing 1 if $y_0\in \mathbb {Z}$ or 0 if $y_0\notin \mathbb {Z}$, and continuing for all $T_n,\ n\ge 1$, by writing

  • 0 if $x_n\in \mathbb {Z}$ and $y_n\notin \mathbb {Z}$;

  • 1 if $y_n\in \mathbb {Z}$ and $x_n\notin \mathbb {Z}$;

  • 01 if $x_n,y_n\in \mathbb {Z}$.

For the previous definitions, the reader can also see (Lothaire 2002, Remark 2.1.12). We now give a very easy example.

Example 3

Considering the line $y=2x/5$, we trivially find

$$\begin{aligned} K_{2/5,\,0}\, =\, 0001001\ 0001001\ 0001001\ \ldots \end{aligned}$$

(6)

Hence, the lower cutting sequence relative to the line $y=2x/5$ is an infinite periodic word with period ⁴ 7 and periodic factor 0001001.

We are now ready to give the central definition of the section. Recalling the construction of the Carboncettus octagon $C_n$ in the previous section, let

$$\begin{aligned} y\,=\,\beta _n x \end{aligned}$$

(7)

be the line through the origin and the point $B_n,\ n\ge 1$ (Fig. 3).

Definition 2

(The Carboncettus word $W_n$) For every $n\in \mathbb {N}$, we set

$$\begin{aligned} W_n\ :=\ K_{\beta _n,0} \end{aligned}$$

and we call $W_n$ the nth Carboncettus infinite word or, simply, the nth Carboncettus word.

Remark 2

Looking at Fig. 3, we can easily note that

$$\begin{aligned} |OH_n| \,=\, \sqrt{|OB_n|^2-|H_nB_n|^2} \,=\, \sqrt{ F_{n+2}^2 - F_n^2 }\,; \end{aligned}$$

hence,

$$\begin{aligned} \beta _n \,=\, \tan (\measuredangle H_nOB_n) \,=\, \frac{F_n}{\sqrt{F_{n+2}^2-F_n^2}} \end{aligned}$$

(8)

for all $n\in \mathbb {N}$.

Example 4

(The first Carboncettus word $W_1$) From Definition 2, the first Carboncettus word is the lower cutting sequence relative to the line obtained as extension of the segment $OB_1$ in Fig. 2. Using (8) or recalling from Example 1 that $\measuredangle H_1 OB_1=\pi /6$, we have

$$\begin{aligned} \beta _1 = \frac{1}{\sqrt{3}} \approx 0.577\,350\,269. \end{aligned}$$

Then, we can easily compute the first digits of $W_1$ as follows:

$$\begin{aligned} W_1\,= & {} K_{ {1}/{\sqrt{3}},\,0} \nonumber \\= & {} 001\,001\,001\, 010\,010\,010\,010\, 100\,100\,100\, \nonumber \\&101\, 001\,001\, 010\,010\,010\,010\, 100\,100\,100\, \nonumber \\&101\, 001\,001\,001\, 010\,010\,010\, 100\,100\,100\, \\&101\, 001\,001\,001\, 010\,010\,010\,010\, 100\,100\, \nonumber \\&101\, 001\,001\,001\, 010\,010\,010\,010\, 100\, \ldots \nonumber \end{aligned}$$

(9)

Example 5

(The Carboncettus words $W_2, W_3$ and $W_4$)

• (i) Using (8), we obtain

  $$\begin{aligned} \beta _2 = \frac{F_2}{\sqrt{F_4^2-F_2^2}}=\frac{\sqrt{2}}{4} \approx 0.353\,553\,390; \end{aligned}$$

  hence, by some simple computations, we find the first digits of the Carboncettus word $W_2$ as follows:

  $$\begin{aligned} W_2\,= & {} K_{\sqrt{2}/4,\,0} \nonumber \\= & {} 000\,100\,010\, 001\,000\,100\,010\,010\,001\, 000\, \nonumber \\&100\, 010\, 001\, 000\, 100\,100\,010\, 001\, 000\,100\, \nonumber \\&010\,001\,001\,000\, 100\,010\, 001\, 000\, 100\, 010\, \\&010\, 001\, 000\, 100\,010\, 001\, 000\, 100\,100\, 010\, \nonumber \\&001\, 000\, 100\, 010\, 010\, 001\, 000\, 100\, 010\,\nonumber \ldots \end{aligned}$$

  (10)

• (ii) Using (8) as before, we find

  $$\begin{aligned} \beta _3 = \frac{F_3}{\sqrt{F_5^2-F_3^2}}=\frac{2\sqrt{21}}{21} \approx 0.436\,435\,780 \end{aligned}$$

  and, by simple computations, we get the first digits of the Carboncettus word $W_3$ as follows:

  $$\begin{aligned} W_3= & {} K_{2\sqrt{21}/21,\,0} \nonumber \\= & {} 000\, 100\,100\,100\, 010\,010\,010\, 001\,001\,001\, \nonumber \\&001\, 000\, 100\,100\,100\, 010\,010\,010\,010\, 001\, \nonumber \\&001\,001\, 000\, 100\,100\,100\,100\, 010\,010\,010\, \\&001\,001\,001\, 000\, 100\,100\,100\,100\, 010\,010\, \nonumber \\&010\, 001\,001\,001\,001\, 000\, 100\,100\,100\,\nonumber \ldots \end{aligned}$$

  (11)

• (iii) Using (8) as in (i) and (ii), we get

  $$\begin{aligned} \beta _4 = \frac{F_4}{\sqrt{F_6^2-F_4^2}}=\frac{3\sqrt{55}}{55} \approx 0.404\,519\,917 \end{aligned}$$

  and then the following first 147 digits of the Carboncettus word $W_4$

  $$\begin{aligned} W_4= & {} K_{3\sqrt{55}/55,\,0} \nonumber \\= & {} 000\, 100\,100\, 010\,010\, 001\,001\, 000\, 100\,100\, \nonumber \\&010\,010\, 001\,001\, 000\, 100\,100\, 010\,010\, 001\, \nonumber \\&001\,001\, 000\, 100\,100\, 010\,010\, 001\,001\, 000\, \\&100\,100\, 010\,010\, 001\,001\, 000\, 100\,100\, 010\, \nonumber \\&010\,010\, 001\,001\, 000\, 100\,100\, 010\,010\,\nonumber \ldots \end{aligned}$$

  (12)

The definition of mechanical word or mechanical sequence is similar to Definition 1. As usual, we let $\lfloor x\rfloor$ and $\lceil x\rceil$ denote, respectively, the floor and the ceiling of a real number x.

Definition 3

For any $\alpha ,\rho \in \mathbb {R}$ with $0\le \alpha \le 1$, we define two infinite words

$$\begin{aligned} s_{\alpha ,\rho }:\mathbb {N}_0\longrightarrow \{0,1\}, \qquad s'_{\alpha ,\rho }:\mathbb {N}_0\longrightarrow \{0,1\} \end{aligned}$$

by setting

$$\begin{aligned} s_{\alpha ,\rho }(n):=\lfloor \alpha (n+1)+\rho \rfloor - \lfloor \alpha n+\rho \rfloor \end{aligned}$$

and

$$\begin{aligned} s'_{\alpha ,\rho }(n):=\lceil \alpha (n+1)+\rho \rceil - \lceil \alpha n+\rho \rceil \end{aligned}$$

for all $n\in \mathbb {N}_0$. The word $s_{\alpha ,\rho }$ [$s'_{\alpha ,\rho }$, respectively] is called the lower [upper, resp.] mechanical word with slope $\alpha$ and intercept $\rho$. Moreover, a mechanical word $s_{\alpha ,\rho }$ or $s'_{\alpha ,\rho }$ is said rational if $\alpha \in \mathbb {Q}$ and irrational if $\alpha \notin \mathbb {Q}$.

There are several equivalent definitions of Sturmian word related to different properties, for which we refer to (Lothaire 2002, Section 2). The simplest way to define it is the following.

Definition 4

A Sturmian word is an infinite word w over an alphabet A which contains exactly $n+1$ factors of length n for all $n\in \mathbb {N}_0$.

For instance, considering factors of length 1, the definition above implies that every Sturmian word must necessarily be over two letters of A, i.e., a binary word. The following theorem establishes which Carboncettus words are Sturmian.

Theorem 1

All Carboncettus words $W_n$ are Sturmian words except when $n=5$; in this case, we have an infinite purely periodic word with period 17 and periodic factor $000\,100\,100\,010\,010\,01$.

Proof From Theorem 2.1.13 of Lothaire (2002), a lower mechanical word $s_{\alpha ,\rho }$ is a Sturmian word if and only if it is irrational or, equivalently, if and only if the slope $\alpha$ is irrational. The Carboncettus word $W_n$ is equal to $K_{\beta _n,0}$ by definition, and it is well known and easy to prove that a lower cutting sequence $K_{\beta ,\rho }$ is equal to the lower mechanical word obtained by “dividing the couple $(\beta ,\rho )$ by $\beta +1$,” that is, in symbols

$$\begin{aligned} K_{\beta ,\rho } = s_{\beta /(\beta +1),\, \rho /(\beta +1) } \end{aligned}$$

(see, for instance, Lothaire 2002, Section 2.1.2). Since $\beta /(\beta +1)=1-1/(\beta +1)$, then $\beta /(\beta +1)$ is irrational if and only if $\beta$ itself is irrational. In conclusion, we have

$$\begin{aligned} W_n = s_{\beta _n/(\beta _n+1),0} \ \text { is Sturmian} \quad \Leftrightarrow \quad \beta _n\notin \mathbb {Q}. \end{aligned}$$

(13)

Now, from Remark 2 we have

$$\begin{aligned} \displaystyle \beta _n= & {} \displaystyle \frac{F_n}{\sqrt{F_{n+2}^2-F_n^2}} \nonumber \\= & {} \displaystyle \frac{F_n}{\sqrt{(F_{n+2}-F_n)(F_{n+2}+F_n)}} \\= & {} \displaystyle \frac{F_n}{\sqrt{F_{n+1}(F_{n+2}+F_n)}}\nonumber \end{aligned}$$

(14)

for all $n\in \mathbb {N}$. By an induction argument on n, it is immediate to prove that

$$\begin{aligned} F_{n+m}= F_n\cdot F_{m-1} + F_{n+1}\cdot F_m \end{aligned}$$

(15)

for all $n\ge 0,\ m\ge 1$: For this purpose, consider $m\ge 1$ fixed and assume that (15) is true for $n=k$ and $n=k+1$. Adding side to side, (15) is hence true for $n=k+2$, and since it trivially holds for $n=0$ and $n=1$, (15) is proved in general.

Using (15) with $m=n+2$, we then get from (14)

$$\begin{aligned} \beta _n = \frac{F_n}{\sqrt{F_nF_{n+1}+F_{n+1}F_{n+2}}} = \frac{F_n}{\sqrt{F_{2n+2}}} \end{aligned}$$

(16)

for all $n\ge 1$;⁵ hence, $\beta _n$ is rational if and only if $F_{2n+2}$ is a perfect square. But Cohn and Wyler proved independently in 1964 that a Fibonacci number $F_m,\ m\in \mathbb {N}_0$, is a square if and only if $m=0,1,2,12$ (see Cohn 1964 and Wyler and Rollett (1964)); hence,

$$\begin{aligned} F_{2n+2},\ n\ge 1,\ \text {is a square} \quad \Leftrightarrow \quad n=5. \end{aligned}$$

(17)

Therefore, (13), (16) and (17) together prove the first part of the thesis.

If $n=5$, we get from (8)

$$\begin{aligned} \beta _5 =\frac{F_5}{\sqrt{F_{7}^2-F_5^2}} = \frac{5}{12}, \end{aligned}$$

and considering the line $y=5x/12$, by some trivial computations, we conclude that $W_5=K_{5/12,\,0}$ is a purely periodic word with period $5+12=17$ and periodic factor 000 100 100 010 010 01, i.e.,

$$\begin{aligned} W_5= & {} 00010010001001001 \quad 00010010001001001 \nonumber \\&00010010001001001 \quad \ldots \end{aligned}$$

(18)

$\square$

In the next example, we compute the first digits of three more words after the special case $W_5$. We will also need these explicit determinations in Example 8 of the next section.

Example 6

(The words $W_6,\, W_7$ and $W_8$)

• (i) Using (8), we obtain

  $$\begin{aligned} \beta _6 = \frac{F_6}{\sqrt{F_8^2-F_6^2}}= \frac{8\sqrt{377}}{377} \approx 0.412\,020\,962; \end{aligned}$$

  hence, as in Example 5, we find the first 147 digits of the Carboncettus word $W_6$ as follows:

  $$\begin{aligned} W_6= & {} K_{8\sqrt{377}/377,\,0} \nonumber \\= & {} 000\, 100\,100\, 010\,010\, 001\,001\,001\, 000\, 100\, \nonumber \\&100\, 010\,010\, 001\,001\,001\, 000\, 100\,100\, 010\, \nonumber \\&010\, 001\,001\,001\, 000\, 100\,100\, 010\,010\, 001\, \\&001\,001\, 000\, 100\,100\, 010\,010\, 001\,001\,001\, \nonumber \\&000\, 100\,100\, 010\,010\, 001\,001\,001\, 000\,\ldots \nonumber \end{aligned}$$

  (19)

• (ii) For $W_7$, we find

  $$\begin{aligned} \beta _7 = \frac{F_7}{\sqrt{F_9^2-F_7^2}}=\frac{13\sqrt{987}}{987} \approx 0.413\,794\,559 \end{aligned}$$

  and then we get the following first 147 digits

  $$\begin{aligned} W_7= & {} K_{13\sqrt{987}/987,\,0} \nonumber \\= & {} 000\, 100\,100\, 010\,010\, 001\,001\,001\, 000\, 100\, \nonumber \\&100\, 010\,010\,010\, 001\,001\, 000\, 100\,100\, 010\, \nonumber \\&010\,010\, 001\,001\, 000\, 100\,100\,100\, 010\,010\, \\&001\,001\, 000\, 100\,100\,100\, 010\,010\, 001\,001\, \nonumber \\&001\, 000\, 100\,100\, 010\,010\, 001\,001\,001\,\nonumber \ldots \end{aligned}$$

  (20)

• (iii) For $W_8$, we find

  $$\begin{aligned} \beta _8 = \frac{F_8}{\sqrt{F_{10}^2-F_8^2}}=\frac{21\sqrt{646}}{1292} \approx 0.413\,116\,974 \end{aligned}$$

  and, consequently,

  $$\begin{aligned} W_8= & {} K_{21\sqrt{646}/1292,\,0}\nonumber \\= & {} 000\, 100\,100\, 010\,010\, 001\,001\,001\, 000\, 100\, \nonumber \\&100\, 010\,010\, 001\,001\,001\, 000\, 100\,100\, 010\, \nonumber \\&010\,010\, 001\,001\, 000\, 100\,100\, 010\,010\,010\, \nonumber \\&001\,001\, 000\, 100\,100\, 010\,010\,010\, 001\,001\, \\&000\, 100\,100\,100\, 010\,010\, 001\,001\, 000\, 100\, \nonumber \\&100\,100\, 010\,010\, 001\,001\, 000\, 100\,100\,\nonumber \ldots \end{aligned}$$

  (21)

  Note that for $W_8$ we have written 30 more digits (177) than for the previous words: The reason for this will be clear later.

4 The Carboncettus limit word

If A is a finite alphabet, the set of right infinite words over A is usually denoted by $A^{\mathbb {N}_0}$ or $A^{\omega }$. It is equipped with a distance d defined as follows: For any

$$\begin{aligned} x=x_0x_1\ldots x_k\ldots \text {and} y=y_0y_1\ldots y_k\ldots \end{aligned}$$

(22)

belonging to $A^{\omega }$, we set

$$\begin{aligned} d(x,y):={\left\{ \begin{array}{ll}\displaystyle 2^{-\min \left\{ k\in \mathbb {N}_0\,:\,x_k\ne y_k\right\} } &{}\text {if }\, x\ne y; \\ 0 &{}\text {if }\, x=y. \end{array}\right. } \end{aligned}$$

(23)

In this way, $(A^{\omega },d)$ is a compact metric space (hence complete and totally bounded by a fundamental result in general topology) called the Cantor space (see Lothaire 2002, Chap. 1). From the distance defined in (23), it follows that a sequence of words $\left\{ X_n\right\} _n\subset A^{\omega }$ converges to $Y\in A^{\omega }$, and we write

$$\begin{aligned} \lim _{n\rightarrow \infty } X_n\ =\ Y \end{aligned}$$

(24)

in this case, if, for every $i\in \mathbb {N}_0$, we have $(X_n)_i=Y_i$ for all sufficiently large n, i.e., greater than some $\nu (i)$. In agreement with (22), the previous notations

$$\begin{aligned} (X_n)_i \text { and } Y_i \end{aligned}$$

obviously indicate the letter of the words $X_n$ and Y, respectively, associated with i.

Now that we have the notations and definitions necessary for this section, let us reconsider Fibonacci numbers. Recalling that the sequence of ratios of two consecutive Fibonacci numbers $\left\{ F_{n+1}/F_n\right\} _{n\in \mathbb {N}_0}$ converges to the golden section $\phi = \left( 1+\sqrt{5}\right) \big /2$,⁶ then

$$\begin{aligned} \displaystyle \lim _{n\rightarrow \infty } \frac{F_{n+2}^2-F_n^2}{F_n^2}= & {} \displaystyle \lim _{n\rightarrow \infty } \left( \frac{F_{n+2}}{F_n}\right) ^2-1 \nonumber \\= & {} \displaystyle \lim _{n\rightarrow \infty } \left( \frac{F_{n+2}}{F_{n+1}}\cdot \frac{F_{n+1}}{F_n} \right) ^2-1 \nonumber \\= & {} \displaystyle \left( \phi ^2 \right) ^2-1 \nonumber \\= & {} \displaystyle 3\phi +1\,, \end{aligned}$$

(25)

where in the last step we used the equality

$$\begin{aligned} \phi ^2\ =\ \phi +1. \end{aligned}$$

(26)

Recalling (8), we then define

$$\begin{aligned} \beta _{\infty }:= & {} \displaystyle \lim _{n\rightarrow \infty }\beta _n = \lim _{n\rightarrow \infty } \frac{F_n}{\sqrt{F_{n+2}^2-F_n^2}}\nonumber \\\\= & {} \displaystyle \lim _{n\rightarrow \infty } \left( \sqrt{ \frac{F_{n+2}^2-F_n^2}{F_n^2} }\, \right) ^{-1}\nonumber \end{aligned}$$

(27)

and, using (25) and (26), we obtain

$$\begin{aligned} \displaystyle \beta _{\infty }= & {} \displaystyle \left( \sqrt{3\phi +1}\right) ^{-1} = \sqrt{ \frac{1}{3\phi +1}\cdot \frac{3\phi -4}{3\phi -4} } \nonumber \\= & {} \displaystyle \sqrt{ \frac{3\phi -4}{5} } = \sqrt{ \frac{3\sqrt{5}-5}{10} } \\\approx & {} 0.4133042381.\nonumber \end{aligned}$$

(28)

This leads to consider the line

$$\begin{aligned} y\,=\,\beta _{\infty } x \end{aligned}$$

(29)

and to state the following:

Definition 5

Similar to Definition 2, we set

$$\begin{aligned} W_{\infty } := K_{\beta _{\infty },0} = K_{\sqrt{(3\sqrt{5} -5)/10}, \ 0} \end{aligned}$$

(30)

and we call $W_{\infty }$ the Carboncettus limit word.

The first digits of $W_{\infty }$ can be easily computed as in Examples 4, 5 and 6, getting

$$\begin{aligned} W_{\infty }= & {} K_{\sqrt{(3\sqrt{5} -5)/10}, \ 0} \nonumber \\= & {} 000\, 100\,100\, 010\,010\, 001\,001\,001\, 000\, 100\, \nonumber \\&100\, 010\,010\, 001\,001\,001\, 000\, 100\,100\, 010\, \nonumber \\&010\,010\, 001\,001\, 000\, 100\,100\, 010\,010\,010\, \nonumber \\&001\,001\, 000\, 100\,100\, 010\,010\,010\, 001\,001\, \\&000\, 100\,100\,100\, 010\,010\, 001\,001\, 000\, 100\, \nonumber \\&100\,100\, 010\,010\, 001\,001\,001\, 000\, 100\,\nonumber \ldots \end{aligned}$$

(31)

Consider now a family, or better, a sequence of lines $\{L_n:n\in \mathbb {N}\}$ with positive slopes in the plane $\mathbb {R}^2$, and assume that it “converges” to some line L of the form (5). Then, it is not true in general that the cutting sequence (lower or upper) attached to $L_n$ approaches the cutting sequence attached to L in the sense of the definition given in (24). And the same can be said for the slopes of a sequence of mechanical or balanced words. The following is a very simple example of what can happen.

Example 7

For any $n\in \mathbb {N}$, consider the line

$$\begin{aligned} L_n:\ \ y=\left( \frac{2}{5}-\frac{1}{n}\right) x \end{aligned}$$

through the origin of the plane. It is clear that the sequence $\left\{ L_n\right\} _{n\in \mathbb {N}}$ converges (punctually) to the line $L: y=2x/5$ considered in Example 3, but the sequence of words $\left\{ K_{2/5-1/n,\,0} \right\} _{n\in \mathbb {N}}$ does not converge to $K_{2/5,\,0}$  shown in (6). In fact, it is simple to check that, for every $n\ge 15$, the first nine digits of $K_{2/5-1/n,\,0}$ are $000\,100\,010$, hence different from the ones of $K_{2/5,\,0}$.

We are now ready to prove the main result of the section.

Theorem 2

The limit of the sequence of Carboncettus words is the Carboncettus limit word, in symbols

$$\begin{aligned} \lim _{n\rightarrow \infty } W_n = W_{\infty }. \end{aligned}$$

Proof

We want to show that for each fixed $i\in \mathbb {N}_0$ it holds

$$\begin{aligned} (W_n)_i = (W_{\infty })_i \end{aligned}$$

(32)

for all $n\in \mathbb {N}$ large enough. Therefore, assume by reductio ad absurdum that this is not true for all i, and let j be the smallest i such that (32) is not satisfied for infinitely many $n\in \mathbb {N}$. Then, for each positive integer $m\le j$ consider the following unitary open interval

$$\begin{aligned} \big ]\lfloor \beta _{\infty }\cdot m\rfloor ,\, \lceil \beta _{\infty }\cdot m\rceil \big [\,, \end{aligned}$$

and since $\beta _n\rightarrow \beta _{\infty }$ by definition and $\beta _{\infty }\notin \mathbb {Q}$ (recall (27) and (28), resp.), then there exists $\nu \in \mathbb {N}$ such that

$$\begin{aligned} \beta _n\cdot m\ \in \ \big ]\lfloor \beta _{\infty }\cdot m\rfloor ,\, \lceil \beta _{\infty }\cdot m\rceil \big [ \end{aligned}$$

for all $n\ge \nu$ and $m\le j$. Now, recalling the definition of lower cutting sequence (see Definition 1(i)), this means that for every integer i from 0 to at least j we have $(W_n)_i=(W_{\infty })_i$ for all $n\ge \nu$, and this contradicts our assumption. $\square$

The previous theorem claims that the sequence of words whose first elements are (9)–(12), (18) and (19)–(21) converges to (31). Using the definition of distance given in (23), it is possible to define a further sequence

$$\begin{aligned} \left\{ \delta _n\right\} _{n\in \mathbb {N}} \quad \text { where } \quad \delta _n\,:=\,d(W_n,W_{\infty }). \end{aligned}$$

(33)

Example 8

Numerically, the first elements of $\left\{ \delta _n\right\} _n$ are the following:

$$\begin{aligned} \displaystyle \delta _1= & {} 2^{-2}, \quad \displaystyle \delta _2\ =\ 2^{-6}, \nonumber \\ \displaystyle \delta _3= & {} 2^{-9}, \quad \displaystyle \delta _4\ =\ 2^{-23}, \\ \displaystyle \delta _5= & {} 2^{-16}, \quad \displaystyle \delta _6\ =\ 2^{-64}, \nonumber \\ \displaystyle \delta _7= & {} 2^{-40}, \quad \displaystyle \delta _8\ =\ 2^{-170},\nonumber \end{aligned}$$

(34)

as the reader can immediately verify comparing (9)–(12) and (18)–(21) with (31).

5 Conclusion and future work

After the introductive section, to move faster to present Carboncettus words, we have reduced to a minimum an important geometric section like the second one. In a later work (on “the Prato octagon and regularity measures for polygons,” see Caldarola 2020), we will devote more space to the sequence $\left\{ C_n\right\} _n$, studying in more depth the characteristics of its elements. For example, already in Caldarola et al. (2020c) there was mention of the existence of the “limit normalized octagon” $C_{\infty }^N$, in analogy with $W_{\infty }$. Since the circumferences $\varGamma _{F_n}$ and $\varGamma _{F_{n+2}}$ become larger and larger, $C_{\infty }^N$ is obtained by normalizing the radius of the internal one. Another approach to study the limit of $\left\{ C_n\right\} _n$ without carrying out normalizations, say $C_{\infty }$, is to use non-standard mathematics or a numerical system that allows calculations with infinite numbers.⁷

We find it fascinating to be able to construct regular or almost regular geometric figures, by means of simple or approximate methods, such as the one shown here which uses Fibonacci numbers to obtain a family of octagons that are indistinguishable from a regular one. And this must be understood not only limited to the figures of the Euclidean plane. The interest, moreover, is undoubtedly greater if it is possible to trace ancient uses or historical foundations of approximate methods, very rare until modern times, as far as we know today.

Much work remains to be done also on the purely combinatorial-algebraic aspects highlighted here, such as the sequence of Carboncettus words and its properties, similar or derived sequences from $\left\{ C_n\right\} _n$, the characteristics of the elements in $\left\{ C_n\right\} _n$, etc. For example, the sequence $\left\{ \delta _n\right\} _n$ defined in (33) does not decrease monotonously as it is immediately evident from (34), but it is easy to prove that the two subsequences of odd- and even-indexed terms, i.e., $\left\{ \delta _{2n+1}\right\} _n$ and $\left\{ \delta _{2n}\right\} _n$, respectively, are monotone (this is because they depend on the approach of the line (7) to (29) from the above and below, resp.). Thus, it seems interesting to study and compare their speeds of convergence to zero and other aspects also in perspective of the so-called unimaginable numbers.

The unimaginable numbers are numbers so large that they cannot be written through the ordinary scientific or exponential notation, but they need notational systems specifically designed for the purpose, such as Knuth up-arrow notation, Conway chained arrow notation, Steinhaus–Moser notation, Bowers’s operators and others (see Caldarola et al. 2020b, d and the references therein for more details). The usual convention is that an unimaginable number is a number greater than 1 googol $=10^{100}$.⁸

In a sequence of converging words, it often happens that the distances decrease very quickly and become very soon “unimaginably small.” For instance, for the fourth element of the sequence $\left\{ \delta _{2n}\right\} _n$, corresponding to the fourth line of the form (7) from the above, we already have

$$\begin{aligned} \delta _8=d(W_8,W_{\infty })=2^{-170} \approx \frac{6.682}{10^{52}} < \frac{1}{\sqrt{1\ \text {googol}\,}}, \end{aligned}$$

and, more importantly, note the rapid growth of the exponents in the right column in (34). Unfortunately, however, despite various evidences, many links between combinatorics of words and unimaginable numbers remain almost unexplored nowadays, and we hope that many researchers, from different parts of the world, will soon be interested in systematically studying these fascinating topics.