Regular polyhedra have identical faces and identical vertices. Additionally every face is a regular polygon and every vertex is regular. There are five regular polyhedra, also called Platonic solids: tetrahedron (triangle faces, 3 in each vertex), hexahedron or cube (square faces, 3 in each vertex), octahedron (triangle faces, 4 in each vertex), dodecahedron (pentagonal faces, 3 in each vertex), icosahedron (triangle faces, 5 in each vertex).

Semi-regular polyhedra have identical vertices. Additionally every face is a regular polygon. The semi-regular polyhedra are also called Archimedean solids.

Every regular and semi-regular polyhedron is described by its vertex index. The index is a sequence of integers. Each of them denotes the number of sides of a face in this vertex. For example (4, 6, 6) is the index of a polyhedron with vertices of degree 3 - one square and two hexagons join in each vertex to form a truncated octahedron.

There are great polyhedra pages by Roman E. Maeder at ETH in Zurich (http://www.mathconsult.ch/showroom/unipoly/index.html) and George W. Hart (http://www.georgehart.com).

Tetrahedron |
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Cube (hexahedron) |
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Octahedron |
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Dodecahedron |
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Icosahedron |

Truncated tetrahedron |
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Truncated cube |
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Cubooctahedron |
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Truncated octahedron |
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Snub cube |
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Lesser rhombicubooctahedron |
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Greater rhombicubooctahedron |
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Truncated dodecahedron |
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Icosidodecahedron |
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Truncated icosahedron |
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Snub dodecahedron |
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Lesser rhombicosidodecahedron |
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Great rhombicosidodecahedron |

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