The farfalle mystery

Abstract

In 2017, for the $\pi$-day, Mickaël Launay described a remarkable property of the Fibonacci sequence, which he called “le mystère de la farfalle”. The name “farfalle” refers to a shape of pasta that is famous in France, a butterfly-shaped one, in connection with a likewise shape appearing in the numerical experiments. In these experiments, the Fibonacci numbers are put on the circle modulo $N$, and the segments joining consecutive elements of the sequence are drawn. Launay proposed a prize of $3.14$ Euros to whom will explain the phenomenon he had discovered. We provide such an explanation, which illustrates recent works by Kurlberg and Rudnick and by Bourgain and Glibichuk.