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 ${A}_{5}\times {C}_{2}$.
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  20130322 12:24:17 
Last modified on  20130322 12:24:17 
Owner  mathwizard (128) 
Last modified by  mathwizard (128) 
Numerical id  20 
Author  mathwizard (128) 
Entry type  Definition 
Classification  msc 5100 
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 