R^{-1} = \left( \begin{array}{c}
\pi^{\parallel} \\
\pi^{\perp} 
\end{array} \right),$$ This setup allows the construction of icosahedral quasicrystals via the cut-and-project scheme: $$\begin{aligned}
E^{\parallel}\;  \stackrel{\pi^{\parallel}}{\longleftarrow} \; E^{\parallel} & \oplus E^{\perp} \; \stackrel{\pi^{\perp}}{\longrightarrow}\; E^{\perp} \\  
& \cup \\
& {\mathcal{L}}\end{aligned}$$ where ${\mathcal{L}}$ is one of the hypercubic lattices in ${\mathbb{R}}^6$. Specifically, we follow [@senechal] and consider as window $W$ the projection into the orthogonal space of the Voronoi cell $\mathcal{V}(\mathbf{0})$ of the origin, i.e. $W = \pi^{\perp}(\mathcal{V}(\mathbf{0}))$. Then the model set (cf. [@moody]) $$\label{model_set_ico}
\Sigma(W) := \left\{ \pi^{\parallel}({\boldsymbol{v}}) : {\boldsymbol{v}}\in {\mathcal{L}}, \; \pi^{\perp}({\boldsymbol{v}}) \in W \right\}$$ defines a quasicrystal in ${\mathbb{R}}^3$ with icosahedral symmetry [@senechal; @zappa1; @paolo]; in a similar way, icosahedral tilings of the space can be obtained with the dualisation method [@senechal; @david; @paolo].

In order to define structural transitions between icosahedral model sets with the Schur rotation method, we consider the classification of the crystallographic representations of ${\mathcal{I}}$ embedded in $B_6$ and the subsequent subgroup structure analysis provided by [@zappa1]. Specifically, the crystallographic representations of ${\mathcal{I}}$ form a unique conjugacy class of subgroups of $B_6$, which we denote by ${\mathcal{C}}_{B_6}({\widetilde{\mathcal{I}}})$, whose order is 192, and whose representative is given by . Let ${\mathcal{G}}$ be a maximal subgroup of the 