Properties of the Ammann-Beenker tiling and its square approximants

TL;DR

This paper provides a comprehensive guide for constructing and analyzing the Ammann-Beenker quasicrystal tiling, facilitating studies of its geometrical properties and approximants for researchers in quasiperiodic systems.

Contribution

It offers a detailed, accessible methodology for generating and studying the Ammann-Beenker tiling and its approximants, including new relations and illustrative examples.

Findings

Analytical methods for local environment enumeration

Simplified inflation and deflation transformations

New relations in tiling properties

Abstract

Our understanding of physical properties of quasicrystals owes a great deal to studies of tight-binding models constructed on quasiperiodic tilings. Among the large number of possible quasiperiodic structures, two dimensional tilings are of particular importance -- in their own right, but also for information regarding properties of three dimensional systems. We provide here a users manual for those wishing to construct and study physical properties of the 8-fold Ammann-Beenker quasicrystal, a good starting point for investigations of two dimensional quasiperiodic systems. This tiling has a relatively straightforward construction. Thus, geometrical properties such as the type and number of local environments can be readily found by simple analytical computations. Transformations of sites under discrete scale changes -- called inflations and deflations -- are easier to establish compared to the celebrated Penrose tiling, for example. We have aimed to describe the methodology with a minimum of technicalities but in sufficient detail so as to enable non-specialists to generate quasiperiodic tilings and periodic approximants, with or without disorder. The discussion of properties includes some relations not previously published, and examples with figures.