An "n"-sided antiprism is a
polyhedroncomposed of two parallel copies of some particular "n"-sided polygon, connected by an alternating band of triangles. Antiprisms are a subclass of the prismatoids.
Antiprisms are similar to prisms except the bases are twisted relative to each other, and that the side faces are triangles, rather than quadrilaterials.
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, 2"n" isosceles triangles as faces.
A uniform antiprism has, apart from the base faces, 2"n" equilateral triangles as faces. They form an infinite series of vertex-uniform polyhedra, as do the uniform prisms. For "n"=2 we have as degenerate case the regular
tetrahedron, and for "n"=3 the non-degenerate regular octahedron.
Cartesian coordinatesfor the vertices of a right antiprism with "n"-gonal bases and isosceles triangles are: with "k" ranging from 0 to 2"n"-1; if the triangles are equilateral,:.
symmetry groupof a right "n"-sided antiprism with regular base and isosceles side faces is "Dnd" of order 4"n", 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 Oh of order 48, which has four versions of "D3d" as subgroups.
The symmetry group contains inversion
if and only if"n" is odd.
rotation groupis "Dn" of order 2"n", 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.
Prismatic uniform polyhedron
**triangular antiprism (
** GlossaryForHyperspace | anchor=Prismatic | title=Prismatic polytopes
* [http://home.comcast.net/~tpgettys/nonconvexprisms.html Nonconvex Prisms and Antiprisms]
* [http://www.software3d.com/Prisms.php Paper models of prisms and antiprisms]
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