The computation of overlap coincidence in Taylor-Socolar substitution tiling

TL;DR

This paper demonstrates that the Taylor-Socolar substitution tiling exhibits overlap coincidence, confirming its pure point spectrum and quasicrystalline structure through an algorithmic approach.

Contribution

It applies an existing algorithm to verify overlap coincidence in the Taylor-Socolar tiling, establishing its pure point spectrum and quasicrystalline nature.

Findings

The tiling has overlap coincidence.

The tiling's spectrum is pure point.

The tiling exhibits quasicrystalline structure.

Abstract

Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of \cite{AL} to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure.