Antiprism

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Set of uniform antiprisms

Type	uniform polyhedron
Faces	2 n-gons, 2n triangles
Edges	4n
Vertices	2n
Vertex configuration	3.3.3.n
Symmetry group	D_nd
Dual polyhedron	trapezohedron
Properties	convex, semi-regular vertex-uniform

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

[edit] Forms

(linear antiprism) Tetrahedron - 4 triangles - self dual
(trigonal antiprism) Octahedron - 8 triangles - dual cube
Square antiprism - 8 triangles, 2 squares - dual tetragonal trapezohedron
Pentagonal antiprism - 10 triangles, 2 pentagons - dual pentagonal trapezohedron
Hexagonal antiprism - 12 triangles, 2 hexagons - dual hexagonal trapezohedron
Heptagonal antiprism - 14 triangles, 2 septagons - dual heptagonal trapezohedron
Octagonal antiprism - 16 triangles, 2 octagons - dual octagonal trapezohedron
...
Decagonal antiprism - 20 triangles, 2 decagons - dual decagonal trapezohedron
...
Dodecagonal antiprism - 24 triangles, 2 dodecagons - dual dodecagonal trapezohedron
...
n-gonal antiprism - 2n triangles, 2 n-gons - dual 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.

[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_d 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.