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# Antiprism

In geometry, 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. 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 quadrilaterals.

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. As faces, it has the two n-gonal bases and, connecting those bases, 2n isosceles triangles.

Uniform antiprism

A uniform antiprism has, apart from the base faces, 2n equilateral triangles as faces. As a class, the uniform antiprisms 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.

The dual polyhedra of the antiprisms are the trapezohedra. Their existence was first discussed and their name was coined by Johannes Kepler.

Cartesian coordinates

Cartesian coordinates for the vertices of a right antiprism with n-gonal bases and isosceles triangles are

\( \left( \cos\frac{k\pi}{n}, \sin\frac{k\pi}{n}, (-1)^k h \right) \)

with k ranging from 0 to 2n−1; if the triangles are equilateral,

\( 2h^2=\cos\frac{\pi}{n}-\cos\frac{2\pi}{n}. \)

Volume and surface area

Let a be the edge-length of a uniform antiprism. Then the volume is

\( V = \frac{n \sqrt{4\cos^2\frac{\pi}{2n}-1}\sin \frac{3\pi}{2n} }{12\sin^2\frac{\pi}{n}} \; a^3 \)

and the surface area is

\( A = \frac{n}{2} ( \cot{\frac{\pi}{n}} + \sqrt{3}) a^2. \)

Symmetry

The symmetry group of a right *n*-sided antiprism with regular base and isosceles side faces is D_{nd} of order 4*n*, except in the case of a tetrahedron, which has the larger symmetry group T_{d} of order 24, which has three versions of D_{2d} as subgroups, and the octahedron, which has the larger symmetry group O_{h} of order 48, which has four versions of D_{3d} as subgroups.

The symmetry group contains inversion if and only if *n* is odd.

The rotation group is D_{n} 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 D_{2} as subgroups, and the octahedron, which has the larger rotation group O of order 24, which has four versions of D_{3} as subgroups.

See also

Prismatic uniform polyhedron

Triangular antiprism (Octahedron)

Square antiprism

Pentagonal antiprism

Hexagonal antiprism

Octagonal antiprism

Decagonal antiprism

Dodecagonal antiprism

Apeirogonal antiprism

Grand antiprism – a four dimensional polytope

External links

Weisstein, Eric W., "Antiprism" from MathWorld.

Olshevsky, George, Antiprism at Glossary for Hyperspace.

Olshevsky, George, Prismatic polytopes at Glossary for Hyperspace.

Nonconvex Prisms and Antiprisms

Paper models of prisms and antiprisms

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