New algebraic and geometric constructs arising from Fibonacci numbers

Abstract

AbstractFibonacci numbers are the basis of a new geometric construction that leads to the definition of a family $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ 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 $$\{C_n:n\in \mathbb {N}\}$$ { C n : n ∈ N } and therefore from Fibonacci numbers: From each Carboncettus octagon $$C_n$$ C n , it is possible to obtain an infinite (right) word $$W_n$$ W n on the binary alphabet $$\{0,1\}$$ { 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 $$n=5$$ n = 5 . The fifth Carboncettus word $$W_5$$ 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 $$W_{\infty }$$ W ∞ named the Carboncettus limit word and, as second main result, we prove that the limit of the sequence of Carboncettus words is $$W_{\infty }$$ W ∞ itself.