in the case of transitions between icosahedral tilings of the space. In this work we demonstrate analytically that the Schur rotation is equivalent to a Bain strain transformation between two congruent six-dimensional lattices. We argue that the advantage of the approach discussed here is the possibilty of classifying the boundary conditions and identifying the order parameters of the transition, while the Bain strain allows only a parameterization of the transitions in terms of paths in the centralisers of the maximal subgroups. Therefore, this work paves the way for a dynamical analysis of the possible transition pathways, by constructing Hamiltonians invariant under the symmetry group of the system in terms of the order parameters of the transition and analysing the resulting energy landscapes, as in previous works [@zappa].

The paper is organised as follows. After reviewing, in Section \[embedding\], the crystallographic embedding of the icosahedral group, based on [@zappa1], we provide, in Section \[schur\], the theoretical framework for the analysis of transitions of quasicrystals and point arrays with icosahedral symmetry, by applying the Schur rotation method and generalising it for any maximal subgroup ${\mathcal{G}}$ of the icosahedral group ${\mathcal{I}}$ . In Section \[schur\_comp\] we explicitly compute the Schur rotations for the 