This SymMath notebook goes with my Platonic Solid worksheet.

$n =$ dimension of manifold which our shape resides in.
$\tilde{T}_i \in \mathbb{R}^{n \times n} =$ i'th isomorphic transform in the minimal set.
$\tilde{\textbf{T}} = \{ 1 \le i \le p, \tilde{T}_i \} =$ minimum set of isomorphic transforms that can be used to recreate all isomorphic transforms.
$p = |\tilde{\textbf{T}}| =$ the number of minimal isomorphic transforms.
$T_i \in \mathbb{R}^{n \times n} =$ i'th isomorphic transform in the set of all unique transforms.
$\textbf{T} = \{ T_i \} = \{ 1 \le k, i_1, ..., i_k \in [1,m], \tilde{T}_{i_1} \cdot ... \cdot \tilde{T}_{i_k} \} =$ set of all unique isomorphic transforms.
$q = |\textbf{T}| =$ the number of unique isomorphic transforms.
$v_1 \in \mathbb{R}^n =$ some arbitrary initial vertex.
$\textbf{v} = \{v_i \} = \{ T_i \cdot v_1 \} =$ the set of all vertices.
$m = |\textbf{v}| =$ the number of vertices.
(Notice that $m \le q$, i.e. the number of vertices is $\le$ the number of unique isomorphic transforms.)
$V \in \mathbb{R}^{n \times m}=$ matrix with column vectors the set of all vertices, such that $V_{ij} = (v_j)_i$.
$P_i \in \mathbb{R}^{m \times m} =$ permutation transform of vertices corresponding with i'th transformation, such that $T_i V = V P_i$.

3D:
Tetrahedron (3 dim), dual to Tetrahedron
Cube (3 dim), dual to Octahedron
Octahedron (3 dim), dual to Cube
Dodecahedron (3 dim), dual to Icosahedron
Icosahedron (3 dim), dual to Dodecahedron

4D:
5-cell (4 dim), dual to 5-cell
8-cell (4 dim), dual to 16-cell
16-cell (4 dim), dual to 8-cell
24-cell (4 dim), dual to 24-cell
120-cell (4 dim), dual to 600-cell vertexes, vertex inner products vertex multiplication table
600-cell (4 dim), dual to 120-cell vertexes, vertex inner products vertex multiplication table
120-cell and 600-cell transform pages: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20
120-cell and 600-cell transform multiplication table pages: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20