Polyhedron

From Citizendium, the Citizens' Compendium

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:

regular tetrahedron:4 triangle faces, 4 vertices, 6 edges	cube: 6 square faces, 8 vertices, 12 edges	regular octahedron: 8 triangle faces, 6 vertices, 12 edges	regular dodecahedron: 12 pentagon faces, 20 vertices, 30 edges
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 Archimedian solid. There are 13 Archimedian solids, shown below:

truncated tetrahedron: 4 hexagon + 4 triangle faces 12 vertices, 18 edges	truncated cube: 6 octagon + 8 triangle faces 24 vertices, 36 edges	truncated octahedron: 8 hexagon + 6 square faces 24 vertices, 36 edges	truncated dodecahedron: 12 decagon + 20 triangular faces 60 vertices, 90 edges
truncated icosahedron: 20 hexagon + 12 pentagon faces 60 vertices, 90 edges	cuboctahedron: 8 triangle + 6 square faces 12 vertices, 24 edges	rhombicuboctahedron: 18 square + 8 triangle faces 24 vertices, 48 edges	truncated cuboctahedron (or great rhombicuboctahedron): 8 octagon + 8 hexagon + 12 square faces 48 vertices, 72 edges
icosidodecahedron: 20 triangle + 12 pentagon faces 30 vertices, 60 edges	rhombicosidodecahedron: 20 triangle + 30 square + 12 pentagon faces 60 vertices, 120 edges	truncated icosidodecahedron (or great rhombicosidodecahedron): 30 square + 20 hexagon + 12 decagon faces 120 vertices, 180 edges	snub cube: 32 triangle + 6 square faces 24 vertices, 60 edges
snub dodecahedron: 80 triangle + 12 pentagon faces 60 vertices, 150 edges