Antiprism

Set of uniform antiprisms

Type	uniform polyhedron
Faces	2 n-gons, 2n triangles
Edges	4n
Vertices	2n
Vertex configuration	3.3.3.n
Schläfli symbol	h_0,1{2,2n} s{2,n}
Coxeter-Dynkin diagrams
Symmetry group	D_nd, [2⁺,2n], (2*n), order 4n D_n, [2,n]⁺, (22n), order 2n
Dual polyhedron	trapezohedron
Properties	convex, semi-regular vertex-transitive
Net

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.

1 Uniform antiprism
2 Cartesian coordinates
3 Volume and surface area
4 Symmetry
5 See also
6 External links

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

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

Symmetry

The symmetry group of a right n-sided antiprism with regular base and isosceles side faces is D_nd of order 4n, 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 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.

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

Polyhedron navigator

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Catalan solids (Dual semiregular)	triakis tetrahedron · rhombic dodecahedron · triakis octahedron · tetrakis cube · deltoidal icositetrahedron · disdyakis dodecahedron · pentagonal icositetrahedron · rhombic triacontahedron · triakis icosahedron · pentakis dodecahedron · deltoidal hexecontahedron · disdyakis triacontahedron · pentagonal hexecontahedron

Dihedral regular	dihedron · hosohedron

Dihedral uniform	prisms · antiprisms

Duals of dihedral uniform	bipyramids · trapezohedra

Dihedral others	pyramids · truncated trapezohedra · gyroelongated bipyramid · cupola · bicupola · pyramidal frusta

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