Algorithm for Generating Quasiperiodic Packings of Multi-Shell Clusters

TL;DR

This paper introduces an efficient mathematical approach and algorithm for generating high-dimensional quasiperiodic packings of multi-shell clusters, significantly improving computational feasibility for complex quasicrystal models.

Contribution

The authors develop mathematical results enabling direct generation of quasiperiodic packings in high-dimensional spaces with reduced computational complexity.

Findings

Efficient computation of determinants for 2D and 3D clusters regardless of superspace dimension.

Successful generation of complex 3-shell icosahedral cluster packings in 31-dimensional space.

The computer program can produce hundreds of points in minutes on a personal computer.

Abstract

Many of the mathematical models used in quasicrystal physics are based on tilings of the plane or space obtained by using strip projection method in a superspace of dimension four, five or six. We present some mathematical results which allow one to use this very elegant method in spaces of dimension much higher and to generate directly quasiperiodic packings of multi-shell clusters. We show that in the case of a two-dimensional (resp. three-dimensional) cluster we have to compute only determinants of order three (resp. four), independently of the dimension of the superspace we use. The computer program based on our mathematical results is very efficient. For example, we can easily generate quasiperiodic packings of three-shell icosahedral clusters (icosahedron + dodecahedron + icosidodecahedron) by using strip projection method in a 31-dimensional space (hundreds of points are obtained in a few minutes on a personal computer).