Tilings with Infinite Local Complexity and n-Fold Rotational Symmetry, n=13,17,21

Abstract

A tiling is said to have infinite local complexity (ILC) if it contains infinitely many two-tile patches up to rigid motions. In this work, we provide examples of substitution rules that generate tilings with ILC. The proof relies on Danzer's algorithm, which assumes that the substitution factor is non-Pisot. In addition to ILC, the tiling space of each substitution rule contains a tiling that exhibits (global) n-fold rotational symmetry, n=13,17,21.