Fibonacci words, hyperbolic tilings and grossone

•The theory of grossone is used to study infinite Fionacci words.•The use of the heptagrid, a tiling of the hyperbolic plane, allows us to illustrate these families.•We obtain richer families and more precise descriptions than in the classical theory.

In this paper, we study the contribution of the theory of grossone to the study of infinite Fibonacci words, combining this tool with the help of a particular tiling of the hyperbolic plane: the tiling {7,3}{7,3}, called the heptagrid. With the help of the numeral system based on grossone, we obtain a richer family of infinite Fibonacci words compared with the traditional approach.