Abstract
Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family {C_n:nin 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:nin 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.
Highlights
We associate with each Fibonacci number Fn, n ≥ 1, a geometric construct Cn and an algebraic object Wn, obtaining simultaneously three sequences {Fn}n, {Cn}n and {Wn}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
The main novelty is represented by the definition of an algebraic object for each octagon Cn and, for each Fibonacci number Fn, it is the right infinite word Wn 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 Cn and will be named the nth Carboncettus word
The Carboncettus word Wn is equal to Kβn,0 by definition, and it is well known and easy to prove that a lower cutting sequence Kβ,ρ is equal to the lower mechanical word obtained by “dividing the couple (β, ρ) by β + 1,” that is, in symbols
Summary
We associate with each Fibonacci number Fn, n ≥ 1, a geometric construct Cn and an algebraic object Wn, obtaining simultaneously three sequences {Fn}n, {Cn}n and {Wn}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. Pirillo recently noted that if the radii of the circumferences are equal to two Fibonacci numbers with indices of the same parity and consecutive, 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.. The main novelty is represented by the definition of an algebraic object for each octagon Cn and, for each Fibonacci number Fn, it is the right infinite word Wn on the binary alphabet {0, 1} that will be precisely defined in Sect. There is much to study and investigate on the Carboncettus family of octagons {Cn : n ∈ N}, but we will do it elsewhere because, as anticipated in the Introduction, we must move on to the section to introduce new algebraic constructs
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