The farfalle mystery

In 2017, for the π -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.