$S^1$, hence it is a compact and connected Lie group. Therefore, the angle $\beta \in S^1$ can be chosen as an order parameter for the transitions with tetrahedral symmetry.

In order to compute the Schur rotations between icosahedral quasicrystals with ${\mathcal{T}}$-symmetry, we need to fix the boundary conditions, i.e. imposing the end and the start of the transition to have icosahedral symmetry. In particular, we consider the crystallographic representation ${\widetilde{\mathcal{I}}}_{{\mathcal{T}}}$ of ${\mathcal{I}}$ with the property that ${\widetilde{\mathcal{T}}}= {\widetilde{\mathcal{I}}}\cap {\widetilde{\mathcal{I}}}_{{\mathcal{T}}}$: Given $M_{{\mathcal{T}}}(\beta) \in {\mathcal{Z}}({\widetilde{\mathcal{T}}},{\mathbb{R}}) \cap SO(6)$, we consider the matrix $R_{{\mathcal{T}}}(\beta) := M_{{\mathcal{T}}}(\beta) R \in O(6)$ and impose $$\label{eq_dec}
R_{{\mathcal{T}}}(\beta)^{-1} {\widetilde{\mathcal{I}}}_{{\mathcal{T}}} R_{{\mathcal{T}}}(\beta) \simeq T_1\oplus T_2 \,.$$ We solve (\[eq\_dec\]) with respect to $\beta$; in other words, we look for angles $\hat{\beta} \in S^1$ such that the corresponding matrix $R_{{\mathcal{T}}}(\hat{\beta})$ decomposes into irreps of the representation ${\widetilde{\mathcal{I}}}_{{\mathcal{T}}}$. Specifically, let $M_2$ and $M_3$ denote the generators of ${\widetilde{\mathcal{I}}}_{{\mathcal{T}}}$, and let us define the matrices $K_j(\beta) := R_{{\mathcal{T}}}(\beta)^{-1}M_jR_{{\mathcal{T}}}(\beta)$, for $j = 2,3$. Condition is then equivalent to the following system of $36$ equations: $$\label{36}
\left\{ \begin{aligned}
& \left(K_2(\beta)\right)_{ij} = 0, \; \left(K_2(\beta)\right)_{ji} = 0 \\  
& \left(K_3(\beta)\right)_{ij} = 0, \;  \left(K_3(\beta)\right)_{ji} = 0 \; 
\end{aligned} \right.$$ for $i = 1, 2,3$ and $j = 