The Sequence of Carboncettus Octagons

Abstract

Considering the classic Fibonacci sequence, we present in this paper a geometric sequence attached to it, where the word “geometric” must be understood in a literal sense: for every Fibonacci number $F_n$ we will in fact construct an octagon $C_n$ that we will call the n-th Carboncettus octagon, and in this way we obtain a new sequence $\big \{C_n \big \}_{n}$ consisting not of numbers but of geometric objects. The idea of this sequence draws inspiration from far away, and in particular from a portal visible today in the Cathedral of Prato, supposed work of Carboncettus marmorarius, and even dating back to the century before that of the writing of the Liber Abaci by Leonardo Pisano called Fibonacci (AD 1202). It is also very important to note that, if other future evidences will be found in support to the historical effectiveness of a Carboncettus-like construction, this would mean that Fibonacci numbers were known and used well before 1202. After the presentation of the sequence $\big \{C_n\big \}_{n}$, we will give some numerical examples about the metric characteristics of the first few Carboncettus octagons, and we will also begin to discuss some general and peculiar properties of the new sequence.