On the structure of Ammann A2 tilings

TL;DR

This paper provides a detailed structural analysis of Ammann A2 tilings, proving their self-similarity and demonstrating their lack of robustness, thereby advancing understanding of their geometric and combinatorial properties.

Contribution

The paper establishes a structure theorem for Ammann A2 tilings, showing their inherent self-similarity and non-robustness using novel techniques.

Findings

Ammann A2 tilings are self-similar.

Ammann A2 tilings are not robust.

A structure theorem for Ammann A2 tilings is proved.

Abstract

We establish a structure theorem for the family of Ammann A2 tilings of the plane. Using that theorem we show that every Ammann A2 tiling is self-similar in the sense of [B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete and Computational Geometry 20 (1998) 265-279]. By the same techniques we show that Ammann A2 tilings are not robust in the sense of [B. Durand, A. Romashchenko, A. Shen. Fixed-point tile sets and their applications, Journal of Computer and System Sciences, 78:3 (2012) 731--764].