three maximal subgroups of ${\mathcal{I}}$, namely the tetrahedral group ${\mathcal{T}}$ and the dihedral groups ${\mathcal{D}}_{10}$ and ${\mathcal{D}}_6$, classifying the possible boundary conditions of the transitions, and provide specific examples. We conclude in Section \[conclusions\] by discussing further applications and future work.

Crystallographic embedding of the icosahedral group {#embedding}
===================================================

Let ${\mathcal{L}}$ be a lattice in ${\mathbb{R}}^d$ with generator matrix $B \in GL(d,{\mathbb{R}})$. The point group ${\mathcal{P}}$ of ${\mathcal{L}}$ is the set of all the orthogonal transformations that leave ${\mathcal{L}}$ invariant: $${\mathcal{P}}:= \{ Q \in O(d) : \exists M \in GL(d,{\mathbb{Z}}): QB = BM \}\,.$$ ${\mathcal{P}}$ is finite and does not depend on the matrix $B$ [@zanzotto]. A (finite) group of isometries $G \subseteq O(k)$ is said to be *crystallographic* in ${\mathbb{R}}^k$ if it leaves a lattice ${\mathcal{L}}$ in ${\mathbb{R}}^k$ invariant, or, equivalently, if it is the subgroup of a point group ${\mathcal{P}}\subseteq O(k)$. The crystallographic restriction states that, for $k =2,3$, a crystallographic group $G$ must contain elements of order $1,2,3,4$ or $6$ [@grimm]. Following Levitov and Rhyner [@levitov], we introduce the following:

\[cryst\_def\] Let $G \subseteq O(k)$ be a finite non-crystallographic group of isometries. A *crystallographic representation* of $G$ is a matrix group $\widetilde{G}$ satisfying the following conditions:

1.  