Symmetry properties of Penrose type tilings

TL;DR

This paper explores the symmetry and scaling properties of Penrose tilings, revealing a broader set of scaling factors and infinite inflation centers for most of these factors, which enhances understanding of quasicrystal structures.

Contribution

It demonstrates that the set of scaling factors for Penrose tilings is larger than previously known, with most having infinitely many inflation centers, advancing the mathematical understanding of quasicrystal symmetries.

Findings

The set of scaling factors extends beyond the known powers of $- au$.

Most scaling factors have infinitely many inflation centers.

The results deepen the understanding of the symmetry properties of Penrose tilings.

Abstract

The Penrose tiling is directly related to the atomic structure of certain decagonal quasicrystals and, despite its aperiodicity, is highly symmetric. It is known that the numbers 1, $-\tau $, $(-\tau)^2$, $(-\tau)^3$, ..., where $\tau =(1+\sqrt{5})/2$, are scaling factors of the Penrose tiling. We show that the set of scaling factors is much larger, and for most of them the number of the corresponding inflation centers is infinite.