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

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