Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces

TL;DR

This paper extends the cut and project method for constructing nonperiodic point sets by incorporating p-adic and mixed topologies in internal spaces, broadening the class of quasicrystals studied.

Contribution

It introduces a new framework for model sets with p-adic internal spaces, demonstrating its applicability to well-known tilings like the chair and Robinson square tilings.

Findings

Well-known tilings fit the p-adic model set framework

The scope of cut and project formalism is expanded

Derived the diffractive nature of these tilings

Abstract

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with p-adic topologies or even with mixed Euclidean/p-adic topologies. We show that a number of well known tilings precisely fit this form, including the chair tiling and the Robinson square tilings. Thus the scope of the cut and project formalism is considerably larger than is usually supposed. Applying the powerful consequences of model sets we derive the diffractive nature of these tilings.