Quasicrystal model sets from overlapping self-similar attractors

Abstract

We give a simple computational approach to mathematical quasicrystals, combining cut-and-project methods with self-similarity. Starting with a Pisot unit ${\hat{I}}^2$ and an iterated function system $g_k(z)={\hat{I}}^{2}z+w_k,k=1,...,m$ in a corresponding ring of algebraic integers, we take the attractor $A$ of the conjugate system as an acceptance window. This yields a unique cut-and-project model set $\hat{I}›$ in the complex plane which fulfils $\hat{I}›=\bigcup g_k(\hat{I}›).$ We describe an algorithm which directly determines $\hat{I}›,$ avoiding the difficulties with the fractal structure of $A.$ Classical constructions are based on tiles $A$ of different shape. The present study continues work on models with overlaps, as introduced by Gummelt (1996), Baake and Grimm (2013), Hejda and Pelantová (2016), and Hare, Masáková, and Vávra (2018). In our approach, the overlaps provide a natural decoration of $\hat{I}›.$ The method is illustrated with a variety of pentagonal examples.