A regular polyhedron is a polyhedron such that

• Every face is a regular polygon.
• On each vertex, the same number of edges concur.
• The dihedral angle between any two faces is always the same.

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:

Regular Tetrahedron

It has 6 edges and 4 vertices and 4 faces, each one being an equilateral triangle. Its symmetry group is .

Regular Hexahedron

Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is .

Regular Octahedron

It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is .

Regular Dodecahedron

It has 20 vertices, 30 edges and 12 faces, each one being a regular pentagon. Its symmetry group is .

Regular Icosahedron

It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is .

Figure: The five Platonic solids - created in Blender 2.36. (Download the Blender file for this picture.)

Note: is the alternating group of order , is the symmetric group of order and is the cyclic group with order .