i =1, \ldots, n \right\}.$$ It follows from that the point sets ${\mathcal{C}}_t$ are invariant under the representation ${\mathcal{G}}_1$ of ${\mathcal{G}}$ (cf. ) for all $t \in (0,1)$, and moreover possess icosahedral symmetry for $t=0$ and $t=1$. We refer to [@zappa2] for a detailed method for the construction of finite nested icosahedral point sets via projection, in connection with the structure of viral capsids.

#### Connection with the Bain strain method.

In crystallography and condensed matter physics, the concept of Bain strain relates to deformations of three-dimensional lattices that keep some symmetry preserved, described by a common subgroup of the point groups of the lattices which constitute the initial and final states [@zanzotto]. Indelicato *et al.* [@paolo; @giuliana] provided a higher-dimensional generalisation of the Bain strain for lattices in ${\mathbb{R}}^n$. In this context, given two lattices ${\mathcal{L}}_0$ and ${\mathcal{L}}_1$ with generator matrices $B_0$ and $B_1$, respectively, and a subgroup ${\mathcal{H}}$ of ${\mathcal{P}}({\mathcal{L}}_0)$ and ${\mathcal{P}}({\mathcal{L}}_1)$, a transition between ${\mathcal{L}}_0$ and ${\mathcal{L}}_1$ with symmetry ${\mathcal{H}}$ is a path $B(t) : [0,1] \rightarrow GL(n,{\mathbb{R}})$ such that, if ${\mathcal{L}}_t$ denotes the intermediate lattice with generator matrix $B(t)$, then ${\mathcal{H}}\subseteq {\mathcal{P}}({\mathcal{L}}_t)$, for all $t \in [0,1]$. If ${\mathcal{L}}_0$ and ${\mathcal{L}}_1$ are 6D hypercubic lattices, and 