  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).
