A group theoretical approach to structural transitions of icosahedral quasicrystals and point arrays

TL;DR

This paper introduces a group theoretical method using Schur rotations to analyze and characterize structural transitions in icosahedral quasicrystals and point arrays, applicable to materials science and biology.

Contribution

It extends the Schur rotation concept to aperiodic structures with icosahedral symmetry, providing a purely group theoretical framework for understanding structural transitions.

Findings

Characterization of transitions via group theory

Explicit computations and examples provided

Applicable to finite point sets in chemistry and biology

Abstract

In this paper we describe a group theoretical approach to the study of structural transitions of icosahedral quasicrystals and point arrays. We apply the concept of Schur rotations, originally proposed by Kramer, to the case of aperiodic structures with icosahedral symmetry; these rotations induce a rotation of the physical and orthogonal spaces invariant under the icosahedral group, and hence, via the cut-and-project method, a continuous transformation of the corresponding model sets. We prove that this approach allows for a characterisation of such transitions in a purely group theoretical framework, and provide explicit computations and specific examples. Moreover, we prove that this approach can be used in the case of finite point sets with icosahedral symmetry, which have a wide range of applications in carbon chemistry (fullerenes) and biology (viral capsids).