Antiprism

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Set of uniform antiprisms

Type	uniform polyhedron
Faces	2 p-gons, 2p triangles
Edges	4p
Vertices	2p
Vertex configuration	3.3.3.p
Coxeter-Dynkin diagram
Symmetry group	D_pd
Dual polyhedron	trapezohedron
Properties	convex, semi-regular vertex-transitive

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 quadrilaterials: the vertices are symmetrically staggered.

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.

A uniform antiprism has, apart from the base faces, 2n 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.

1 Cartesian coordinates
2 Symmetry
3 Forms
4 Star antiprisms
5 External links

[edit] Cartesian coordinates

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

$( \cos(k\pi/n), \sin(k\pi/n), (-1)^k a )\;$

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

$2a^2=\cos(\pi/n)-\cos(2\pi/n)\;$ .

[edit] 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.

[edit] Forms

Name	Faces	Dual
Linear antiprism (Tetrahedron)	4 triangles	Tetrahedron
Trigonal antiprism (Octahedron)	8 triangles	Cube
Square antiprism	8 triangles, 2 squares	Tetragonal trapezohedron
Pentagonal antiprism	10 triangles, 2 pentagons	Pentagonal trapezohedron
Hexagonal antiprism	12 triangles, 2 hexagons	Hexagonal trapezohedron
Heptagonal antiprism	14 triangles, 2 heptagons	Heptagonal trapezohedron
Octagonal antiprism	16 triangles, 2 octagons	Octagonal trapezohedron
Nonagonal antiprism	18 triangles, 2 nonagons	Nonagonal trapezohedron
Decagonal antiprism	20 triangles, 2 decagons	Decagonal trapezohedron
Hendecagonal antiprism	22 triangles, 2 hendecagons	Hendecagonal trapezohedron
Dodecagonal antiprism	24 triangles, 2 dodecagons	Dodecagonal trapezohedron
n-gonal antiprism	2n triangles, 2 n-gons	n-gonal trapezohedron

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.