Algorithm for generating quasiperiodic packings of icosahedral three-shell clusters

TL;DR

This paper introduces an efficient algorithm based on the strip projection method for generating quasiperiodic packings of icosahedral three-shell clusters in high-dimensional superspaces, enabling faster computation of complex quasiperiodic structures.

Contribution

The authors develop a highly efficient algorithm for the strip projection method applicable to high-dimensional superspaces, specifically for generating 3D quasiperiodic packings of icosahedral clusters.

Findings

Generated 400-500 points in 10 minutes on a Pentium 4.

Applied the algorithm to 3D icosahedral clusters embedded in 31D superspace.

Demonstrated the algorithm's efficiency for complex quasiperiodic structures.

Abstract

The strip projection method is the most important way to generate quasiperiodic patterns with predefined local structure. We have obtained a very efficient algorithm for this method which allows one to use it in superspaces of very high dimension. A version of this algorithm for two-dimensional clusters and an application to decagonal two-shell clusters (strip projection in a 10-dimensional superspace) has been presented in math-ph/0504036. The program in FORTRAN 90 used in this case is very fast (700-800 points are obtained in 3 minutes). We present an application of our algorithm to three-dimensional clusters. The physical three-dimensional space is embedded into a 31-dimensional superspace and the strip projection method is used in order to generate a quasiperiodic packing of interpenetrating translated copies of a three-shell icosahedral cluster formed by the 12 vertices of a regular icosahedron (the first shell), the 20 vertices of a regular dodecahedron (the second shell) and the 30 vertices of an icosidodecahedron (the third shell). On a personal computer Pentium 4 with Fortran PowerStation version 4.0 (Microsoft Developer Studio) we obtain 400-500 points in 10 minutes. More details, bibliography and samples can be found on the website: http://fpcm5.fizica.unibuc.ro/~ncotfas/