regular polyhedron

A regular polyhedron is a polyhedron such that

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Every face is a regular polygon.
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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: It has 6 edges and 4 vertices and 4 faces, each one being an equilateral triangle. Its symmetry group is $S_{4}$ .
Regular Hexahedron: 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 $A_{5}\\times C_{2}$ .

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.)

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 $n$ .

Title	regular polyhedron
Canonical name	RegularPolyhedron
Date of creation	2013-03-22 12:24:17
Last modified on	2013-03-22 12:24:17
Owner	mathwizard (128)
Last modified by	mathwizard (128)
Numerical id	20
Author	mathwizard (128)
Entry type	Definition
Classification	msc 51-00
Synonym	Platonic solid
Synonym	regular polyhedra
Synonym	regular
Related topic	RegularPolygon
Related topic	Grafix
Defines	tetrahedron
Defines	octahedron
Defines	dodecahedron
Defines	icosahedron
Defines	regular tetrahedron
Defines	regular octahedron
Defines	regular dodecahedron
Defines	regular icosahedron