Polyhedron

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A polyhedron is a three-dimensional geometric closed figure bounded by a connected set of polygons. A polyhedron, in Euclidian geometry, must have at least four faces. A polyhedron of four sides is called a tetrahedron, six sides a hexahedron, eight sides an octahedron, ten sides a decahedron. Figures with more sides are typically named with the Greek name for the number of sides, followed by "-hedron".

The polygons bounding a polyhedron are known as faces; the line segments bounding the polygons are known as edges, and the points where the faces meet are vertices (singular vertex).

A convex polyhedron bounded by faces which are all the same-sized regular polygon is known as a Platonic solid. There are only five Platonic solids, shown below:

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  regular tetrahedron:4 triangle faces, 4 vertices, 6 edges

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  cube: 6 square faces, 8 vertices, 12 edges

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  regular octahedron: 8 triangle faces, 6 vertices, 12 edges

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  regular dodecahedron: 12 pentagon faces, 20 vertices, 30 edges

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  regular icosahedron: 20 triangle faces, 12 vertices, 30 edges

A convex polyhedron bounded by faces of more than one type of regular polygon, with all edges the same length and all vertices identical is called an Archimedean solid. There are 13 Archimedean solids, shown below:

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  truncated tetrahedron:
  4 hexagon + 4 triangle faces
  12 vertices, 18 edges

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  truncated cube:
  6 octagon + 8 triangle faces
  24 vertices, 36 edges

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  truncated octahedron:
  8 hexagon + 6 square faces
  24 vertices, 36 edges

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  truncated dodecahedron:
  12 decagon + 20 triangular faces
  60 vertices, 90 edges

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  truncated icosahedron:
  20 hexagon + 12 pentagon faces
  60 vertices, 90 edges

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  cuboctahedron:
  8 triangle + 6 square faces
  12 vertices, 24 edges

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  rhombicuboctahedron:
  18 square + 8 triangle faces
  24 vertices, 48 edges

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  truncated cuboctahedron (or great rhombicuboctahedron):
  8 octagon + 8 hexagon + 12 square faces
  48 vertices, 72 edges

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  icosidodecahedron:
  20 triangle + 12 pentagon faces
  30 vertices, 60 edges

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  rhombicosidodecahedron:
  20 triangle + 30 square + 12 pentagon faces
  60 vertices, 120 edges

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  truncated icosidodecahedron (or great rhombicosidodecahedron):
  30 square + 20 hexagon + 12 decagon faces
  120 vertices, 180 edges

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  snub cube:
  32 triangle + 6 square faces
  24 vertices, 60 edges

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  snub dodecahedron:
  80 triangle + 12 pentagon faces
  60 vertices, 150 edges