– (XAxisMax); (YAxisMin) – (YAxisMax); in [-5,-4,...,8]{}[ in [-2,-1,...,5]{}[ at (,) ; ]{} ]{}

(-4,-2.472) – (8.4,5.191); (1.5, -2.427) – (-3.5,5.663);

(-2.2,-3.028) – (4.2, 5.78); (3.2,-2.325) – (-5.2, 3.77);

(2, 1.236) to\[bend right\] (1.2, 1.6516);

at (1.5, 1.3) [$\beta$]{};

#### Transitions of finite icosahedral point sets.

In the context of virology and carbon chemistry, the arrangements of viral proteins and carbon atoms in fullerenes are modeled via finite point sets (arrays) with icosahedral symmetry. The method developed here can also be applied to analyse structural transitions between icosahedral arrays, creating finite point sets via projection, as opposed to the infinite ones generated by the cut-and-project scheme, at every time $t$ of the transformation. Indeed, let ${\mathcal{C}}= \{ \pi^{\parallel}({\boldsymbol{v}}_i) : {\boldsymbol{v}}_i \in {\mathcal{L}}, i = 1,\ldots, n \}$ be a finite point set in $E^{\parallel}$, obtained via the projection of points of a hypercubic lattice ${\mathcal{L}}$ in ${\mathbb{R}}^6$. Let us assume that ${\mathcal{C}}$ is closed under the action of the irrep $\rho_3$ of ${\mathcal{I}}$ (cf. ), i.e. $\rho_3 {\mathcal{C}}\subseteq {\mathcal{C}}$. The projection operators $\pi^{\parallel}_t$ given in can be used to define a family of arrays ${\mathcal{C}}_t$, for $t \in [0,1]$, given by: $$\label{clusters}
{\mathcal{C}}_t := \left\{ \pi^{\parallel}_t({\boldsymbol{v}}_i) : {\boldsymbol{v}}_i \in {\mathcal{C}}, 