regular polyhedron (original) (raw)

A regular polyhedron is a polyhedron such that

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  On each vertex, the same number of edges concur.
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  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

Regular Hexahedron

Also known as cube. It has 8 vertices, 12 edges and 6 faces each one being a square. Its symmetry group is S4×C2.

Regular Octahedron

It has 6 vertices, 12 edges and 8 faces, each one being an equilateral triangle Its symmetry group is S4×C2.

Regular Dodecahedron

It has 20 vertices, 30 edges and 12 faces, each one being a regular pentagon. Its symmetry group is A5×C2.

Regular Icosahedron

It has 12 vertices, 30 edges and 20 faces, each one being an equilateral triangle. Its symmetry group is A5×C2.

Figure 1: The five Platonic solids – created in Blender 2.36. (Download the \htmladdnormallinkBlender filehttp://aux.planetmath.org/files/objects/2254/platonic.blend for this picture.)