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

TL;DR

This paper constructs substitution tilings with infinite local complexity that also exhibit global n-fold rotational symmetry for n=13, 17, and 21, expanding understanding of complex symmetric tilings.

Contribution

It introduces new substitution rules generating tilings with infinite local complexity and specific rotational symmetries, using Danzer's algorithm for non-Pisot substitution factors.

Findings

Existence of tilings with ILC and n-fold symmetry for n=13,17,21

Use of Danzer's algorithm for non-Pisot substitution factors

Examples of substitution rules producing these tilings

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.