A guide to lifting aperiodic structures

TL;DR

This paper reviews methods for embedding aperiodic point sets with non-crystallographic symmetry into higher-dimensional spaces, emphasizing Minkowski embedding, and demonstrates its application with a quasicrystal example.

Contribution

It introduces an explicit, number-theoretic approach to embedding aperiodic structures without requiring a lattice, especially useful for quasicrystalline symmetries.

Findings

Effective Minkowski embedding for quasicrystals

Application to a real square-triangle tiling sample

Demonstrates practical utility in physical space analysis

Abstract

The embedding of a given point set with non-crystallographic symmetry into higher-dimensional space is reviewed, with special emphasis on the Minkowski embedding known from number theory. This is a natural choice that does not require an a priori construction of a lattice in relation to a given symmetry group. Instead, some elementary properties of the point set in physical space are used, and explicit methods are described. This approach works particularly well for the standard symmetries encountered in the practical study of quasicrystalline phases. We also demonstrate this with a recent experimental example, taken from a sample with square-triangle tiling structure and (approximate) twelvefold symmetry.