The five regular convex polyhedron in 3-dimensional Cartesian space:
tetrahedron, cube, octahedron, dodecahedron, icosahedron
Regarding a Platonic solid, determined uniquely up to isometry on $\mathbb{R}^3$ as a regular convex polyhedron, as a metric subspace $S$ of $\mathbb{R}^3$. Then is symmetry group may be defined as the group of isometries of $S$.
The groups arising this way are called the groups of ADE-type:
Plato’s Timaeus
The Platonic solids are named after their discussion in
Their construction and the proof that there is exactly five of them appears in
See also