archimedean solid

An Archimedean solid or semiregular solid is a convex polyhedron with regular polygons as faces, such that at least two different types of regular polygons are used, and all vertices are identical (in the sense that the polygons are arranged in the same way about each vertex, and if someone rotates an archimedean solid with no markings randomly, when you aren't looking, it is not possible to work out any information at all about which vertex was moved where.). For short: they are vertex-uniform but not face-uniform. The regular-faced prisms and antiprisms are also semiregular and fit this criteria, but for historical reasons are not included in the definition (probably because there are infinitely many of them, or because of their lesser symmetry). The elongated square gyrobicupola (a Johnson solid) appears at first to meet the above criteria, but upon closer inspection it is not vertex-uniform at all, the vertices being only locally similar. Compare to Platonic solids, which are face-uniform (in addition to being vertex-uniform), and to Johnson solids, which need not be vertex-uniform. The Archimedean solids are known to have been discussed by Archimedes, although the complete record is lost. During the Renaissance, artists and mathematicians valued pure forms and rediscovered all of these forms. This search culminated in the work of Johannes Kepler circa 1619, who defined prisms, antiprisms, and the non-convex solids known as Kepler-Poinsot solids. All edges of an Archimedean solid have the same length, since the faces are regular polygons, and the edges of a regular polygon have the same length. The neighbours of a polygon must have the same edge length, therefore also the neighbours of the neighbours, and so on. There are 13 Archimedean solids (15 if the mirror images of two enantiomorphs, see below, are counted separately).

Name and picture	Faces	Edges	Vertices	Faces meeting at each vertex	Symmetry group
cuboctahedron (Video)	14 (8 triangles, 6 squares)	24	12	triangle-square-triangle-square	O_h
icosidodecahedron (Video)	32 (20 triangles, 12 pentagons)	60	30	triangle-pentagon-triangle-pentagon	I_h
truncated tetrahedron (Video)	8 (4 triangles, 4 hexagons)	18	12	triangle-hexagon-hexagon	T_d
truncated cube or truncated hexahedron (Video)	14 (8 triangles, 6 octagons)	36	24	triangle-octagon-octagon	O_h
truncated octahedron (Video)	14 (6 squares, 8 hexagons)	36	24	square-hexagon-hexagon	O_h
truncated dodecahedron (Video)	32 (20 triangles, 12 decagons)	90	60	triangle-decagon-decagon	I_h
truncated icosahedron or commonly football (soccer ball) (Video)	32 (12 pentagons, 20 hexagons)	90	60	pentagon-hexagon-hexagon	I_h
rhombicuboctahedron or small rhombicuboctahedron (Video)	26 (8 triangles, 18 squares)	48	24	triangle-square-square-square	O_h
truncated cuboctahedron or great rhombicuboctahedron (Video)	26 (12 squares, 8 hexagons, 6 octagons)	72	48	square-hexagon-octagon	O_h
rhombicosidodecahedron or small rhombicosidodecahedron (Video)	62 (20 triangles, 30 squares, 12 pentagons)	120	60	triangle-square-pentagon-square	I_h
truncated icosidodecahedron or great rhombicosidodecahedron (Video)	62 (30 squares, 20 hexagons, 12 decagons)	180	120	square-hexagon-decagon	I_h
snub cube or snub cuboctahedron (2 chiral forms) (Video) (Video)	38 (32 triangles, 6 squares)	60	24	triangle-triangle-triangle-triangle-square	O
snub dodecahedron or snub icosidodecahedron (2 chiral forms) (Video) (Video)	92 (80 triangles, 12 pentagons)	150	60	triangle-triangle-triangle-triangle-pentagon	I

The first two solids (cuboctahedron and icosidodecahedron) are edge-uniform and are called quasi-regular. The last two (snub cube and snub dodecahedron) are known as chiral, as they come in a left-handed (Latin: levomorph or laevomorph) form and right-handed (Latin: dextromorph) form. When something comes in multiple forms which are each other's three-dimensional mirror image, these forms may be called enantiomorphs. (This nomenclature is also used for the forms of chemical compounds). The duals of the Archimedean solids are called the Catalan solids. Together with the bipyramids and trapezohedra, these are the face-uniform solids with regular vertices.

External links

Paper models of Archimedean solids
The Uniform Polyhedra
Virtual Reality Polyhedra The Encyclopedia of Polyhedra
Penultimate Modular Origami

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