New algebraic and geometric constructs arising from Fibonacci numbers

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New algebraic and geometric constructs arising from Fibonacci numbers In honor of Masami Ito Fabio Caldarola1

· Gianfranco d’Atri1 · Mario Maiolo2

· Giuseppe Pirillo3

© The Author(s) 2020

Abstract Fibonacci numbers are the basis of a new geometric construction that leads to the definition of a family {Cn : 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 {Cn : n ∈ N} and therefore from Fibonacci numbers: From each Carboncettus octagon Cn , it is possible to obtain an infinite (right) word Wn 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 W5 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∞ named the Carboncettus limit word and, as second main result, we prove that the limit of the sequence of Carboncettus words is W∞ itself. Keywords Approximate constructions · Computing on words · Fibonacci numbers · Sturmian words · Mechanical sequences · Limit of words · Isogonal polygons

1 Introduction Communicated by Yaroslav D. Sergeyev. M. Ito is one of the world’s leading experts on combinatorial and algebraic properties of words and languages.

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Fabio Caldarola [email protected] Gianfranco d’Atri [email protected] Mario Maiolo [email protected] Giuseppe Pirillo [email protected]

1

Department of Mathematics and Computer Science, Università della Calabria, Cubo 31/A, Arcavacata di Rende, CS 87036, Italy

2

Department of Environmental Engineering, Università della Calabria, Cubo 42/B, Arcavacata di Rende, CS 87036, Italy

3

Department of Mathematics and Computer Science ‘U. Dini’, Università di Firenze, viale Morgagni 67/A, Firenze 50134, Italy

In this paper, we associate with each Fibonacci number Fn , n ≥ 1, a geometric construct Cn and then 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. In particular, {Cn }n is a sequence of octagons very close to a regular one and {Wn }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 app