$n =$ 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 $T_i V = V P_i$.


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