the usual deformation of the classical algebra with a complex root of unity parameter.

Let us discuss a specific model implementation. Most of the geometric quantities of interest such as lengths, areas and volumes can be described in a unifying way and simplest form from the geometry of the 3-simplex, the tetrahedron. The tetrahedron can be described by its 4 (outgoing) normals, vectors $\mathbf{L}_{a}$ in its four triangular faces, $a=1,2,3,4$, subject to the closure constraint $$\mathbf{C}=\sum_{a=1}^{4}\mathbf{L}_{a}=0.\label{eq:classicalclosureconstraint}$$ All geometric properties of the tetrahedron can be derived from the normals and must be invariant under a common $SO(3)$ rotation of the tetrahedron. For example, the area of face $a$ is given by $A_{a}=|\mathbf{L}_{a}|$ and the volume is defined by $V^{2}=\frac{2}{9}\left(\mathbf{L}_{1}\times\mathbf{L}_{2}\right)\cdot\mathbf{L}_{3}$. The proposed quantizing postulate is a quantum deformation of eq. (\[eq:commutationsu2algebra\]). Eq. (\[eq:commutationsu2algebra\]) is the $SU(2)$ algebra implied by the 3 dimensional rotational symmetry, where the normals are promoted to operators $L_{a}$ and identified with the generators of the algebra $L_{a}^{i}$ ($i=1,2,3$). We first write the classical algebra in terms of Cartan generators and root vectors, which are the usual ladder operators of $SU(2)$, then choose the Cartan generator $h_{a}=L_{a}^{3}$ and the root vectors $L_{a}^{\pm\alpha}=L_{a}^{1}\pm\alpha iL_{a}^{2}$, where for $SU(2)$, $\alpha=1$, and we 