Antiprism

An n-sided antiprism is a polyhedron composed of two parallel copies of some particular n-sided polygon, connected by an alternating band of triangles.

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Antiprisms are similar to prisms except the bases are twisted relative to each other: the vertices are symmetrically staggered.

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In the case of a regular n-sided base, one usually considers the case where its copy is twisted by an angle 180?/n. Extra regularity is obtained by the line connecting the base centers being perpendicular to the base planes, making it a right antiprism. It has, apart from the base faces, 2n isosceles triangles as faces.

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A regular right antiprism has, apart from the base faces, 2n equilateral triangles as faces. They form an infinite series of vertex-uniform polyhedra, as do the regular right prisms. For n=2 we have as degenerate case the regular tetrahedron.

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If n=3 then we only have triangles; we get the octahedron, a particular type of right triangular antiprism which is also edge- and face-uniform, and so counts among the Platonic solids. The dual polyhedra of the antiprisms are the trapezohedra. Their existence was first discussed and their name was coined by Johannes Kepler.

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Canonical coordinates for an antiprism with n-gons are (sin(2πk/n), cos(2πk/n), ?a) with k ranging from 0 to n-1, where

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:a=sqrt{rac{cosrac{pi}{n}-cosrac{2pi}{n}}{2}}.

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The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is Dnd of order 4n, except in the case of a tetrahedron, which has the larger symmetry group Td of order 24, which has three versions of D2d as subgroups, and the octahedron, which has the larger symmetry group Od of order 48, which has four versions of D3d as subgroups.

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The symmetry group contains inversion iff n is odd.

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The rotation group is Dn of order 2n, except in the case of a tetrahedron, which has the larger rotation group T of order 12, which has three versions of D2 as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D3 as subgroups.

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Polyhedron: A polyhedron is a geometric shape which in mathematics is defined by three related meanings. In the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. Further gen...

Polygon: :For other use please see Polygon (disambiguation)...

Triangle: :For alternative meanings, such as the musical instrument, see triangle (disambiguation)....