Combinatorial and Computational aspects of Tilings

The aim of this lecture is to discuss the generation of tilings obtained by projection of discretizations of geometric objects such as discrete hyperplanes.

These tilings are dimer models on the hexagonal lattice. We discuss here generation processes for such tilings based
on generalized substitutions, multidimensional continued fractions and S-adic generation. Recall that a substitution is a morphism of the free monoid, which replaces a letter by a word, whereas an $S$-adic sequence is generated as the limit of the composition of an infinite sequence of substitutions with values in a finite set of substitutions. As an application, we discuss convergence to Rauzy fractals of suitable renormalizations of generated patches for some families of parameters (the parametrs are given here by the normal vector of the hyperplanewhich is discretized).