# regular polyhedron

These polyhedra are also known as Platonic solids, since Plato described them in his work. There are only 5 regular polyhedra, as was first shown by Theaetetus, one of Plato’s students. Some sources ascribe to Theaetetus also the discovery of the dodecahedron  .

The five solids are:

Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is $S_{4}\times C_{2}$.

Regular Octahedron

It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is $S_{4}\times C_{2}$.

Regular Dodecahedron

It has 20 vertices, 30 edges and 12 faces, each one being a regular pentagon. Its symmetry group is $A_{5}\times C_{2}$.

Regular Icosahedron

It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is $A_{5}\times C_{2}$.

Note: $A_{n}$ is the alternating group  of order $n$, $S_{n}$ is the symmetric group   of order $n$ and $C_{n}$ is the cyclic group  with order $n$.