lattices in ${\mathbb{R}}^6$ described in Section \[embedding\], and let $\Sigma(W)$ be the icosahedral model set as in . Let us then consider, for all $t \in [0,1]$, the projection $ W_t := \pi^{\parallel}_t((\mathcal{V}(\mathbf{0}))$ of $\mathcal{V}(\mathbf{0})$ into the space $E^{\perp}_t$. We define the family of model sets $$\label{sigma_t}
\Sigma_t \equiv \Sigma(W_t) := \left\{ \pi^{\parallel}_t ({\boldsymbol{v}}) : {\boldsymbol{v}}\in {\mathcal{L}}, \; \pi^{\perp}_t({\boldsymbol{v}}) \in W_t \right\}.$$ By construction, $\Sigma_0 \equiv \Sigma(W)$ and $\Sigma_1$ possess icosahedral symmetry, whereas the intermediate states $\Sigma_t$, for $t \in (0,1)$, display ${\mathcal{G}}$-symmetry since, by : $$\label{comm_G_t}
\pi^{\parallel}_t({\widetilde{\mathcal{G}}}{\boldsymbol{v}}) = {\mathcal{G}}_1 \pi^{\parallel}_t ({\boldsymbol{v}}), \quad \forall t \in (0,1).$$ Hence, the Schur rotation $M_{{\mathcal{G}}}(t)$ as in defines a continuous transformation of the model set $\Sigma(W)$ into another icosahedral quasilattice, where ${\mathcal{G}}$-symmetry is preserved. We point out that, in the higher dimensional space, the lattice ${\mathcal{L}}$ is *fixed* and the transformation is induced by the rotation of the physical and orthogonal spaces (see Figure \[schur\_fig\]). The angle(s) of rotation correspond(s) to the degree(s) of freedom of the transformation, and can be chosen as the *order parameter(s)* of the transition in the framework of the Landau theory [@sethna].

(Origin) at (0,0); (XAxisMin) at (-5,0); (XAxisMax) at (8,0); (YAxisMin) at (0,-2); (YAxisMax) at (0,5); (XAxisMin) 