\right)$

               $Q  = \frac{1}{4} \left( \begin{array}{ccc}  
                         3-\tau & 1 & \tau+2 \\             
                         -\tau-2 & 3-\tau & 1 \\            
                          -1 & -\tau-2 & 3-\tau             
                          \end{array} \right)$              
  --------------------------------------------------------------------------------------------------

  : \[tet\_reps\_table\] Explicit forms of the irreps $\Gamma_1$ and $\Gamma_2$ of the tetrahedral group and the matrix $Q \in O(3)$ such that $Q^{-1} \Gamma_2 Q = \Gamma_1$ (cf. ).

#### Example: tetrahedral transition with an intermediate cubic lattice.

We consider as an explicit example of a tetrahedral transition the path $\beta(t) = \hat{\beta}t$, that connects $0$ with $\hat{\beta} = -\arctan\left(\frac{1}{2}\right)$. The lattice ${\mathcal{L}}$ in ${\mathbb{R}}^6$ is taken as the simple cubic lattice with the standard basis in ${\mathbb{R}}^6$. The matrix $R_{{\mathcal{T}}}(t) = M_{{\mathcal{T}}}(\hat{\beta}t) R$ encodes the projections $\pi^{\parallel}_t$ and $\pi^{\perp}_t$ as in , that define the family of model sets $\Sigma_t$ as in . In Figure \[trans\_tet\] we show a patch of the resulting quasilattices for $t = 0$, $0.5$ and $1$. These are very interesting results; indeed, the starting and final states display icosahedral aperiodicity, as expected by the boundary conditions, while for $t = 0.5$ the corresponding structure is actually a three-dimensional lattice, i.e. it is periodic. Hence such a transition has an intermediate periodic order, which is in accordance to the 