Discrete quasiperiodic sets with predefined local structure

TL;DR

This paper introduces an algorithm to construct discrete quasiperiodic sets with a predefined local structure based on G-clusters, aiding the analysis of quasicrystals' structure and diffraction patterns.

Contribution

The paper presents a novel algorithm that directly constructs multi-component model sets from G-clusters, linking local structure to global quasiperiodic arrangements.

Findings

Algorithm successfully generates quasiperiodic sets with desired local structure.

Potential application in analyzing quasicrystal diffraction patterns.

Provides a mathematical tool for quasicrystal physics research.

Abstract

Model sets play a fundamental role in structure analysis of quasicrystals. The diffraction diagram of a quasicrystal admits as symmetry group a finite group G, and there is a G-cluster C (union of orbits of G) such that the quasicrystal can be regarded as a quasiperiodic packing of interpenetrating copies of C. We present an algorithm which leads from any G-cluster C directly to a multi-component model set Q such that the arithmetic neighbours of any point x belonging to Q are distributed on the sites of the translated copy x+C of C. Our mathematical algorithm may be useful in quasicrystal physics.