{| border="1" bgcolor="#ffffff" cellpadding="5" align="right" style="margin-left:10px" width="250"
!bgcolor=#e7dcc3 colspan=2|Set of uniform antiprisms
|-
|align=center colspan=2|[[image:antiprism17.jpg|240px|Heptadecagonal antiprism]]<br />
|-
|bgcolor=#e7dcc3|Type||[[uniform polyhedron]]
|-
|bgcolor=#e7dcc3|Faces||2 [[polygon|p-gon]]s, 2p [[triangle]]s
|-
|bgcolor=#e7dcc3|Edges||4p
|-
|bgcolor=#e7dcc3|Vertices||2p
|-
|bgcolor=#e7dcc3|[[Vertex configuration]]||3.3.3.p
|-
|bgcolor=#e7dcc3|[[Coxeter-Dynkin diagram]]||[[Image:CDW_hole.png]][[Image:CDW_p.png]][[Image:CDW_hole.png]][[Image:CDW_2b.png]][[Image:CDW_hole.png]]
|-
|bgcolor=#e7dcc3|[[Symmetry group]]||[[Symmetry_group#Three_dimensions|''D''_''pd'']]
|-
|bgcolor=#e7dcc3|[[Dual polyhedron]]||[[trapezohedron]]
|-
|bgcolor=#e7dcc3|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 [[triangle]]s.

Antiprisms are a subclass of the [[prismatoid]]s.

Antiprisms are similar to [[prism (geometry)|prism]]s 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, 2''n'' isosceles triangles as faces.

A '''[[Prismatic uniform polyhedron|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]].

The [[dual polyhedron|dual polyhedra]] of the antiprisms are the [[trapezohedron|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
: <math>( \cos(k\pi/n), \sin(k\pi/n), (-1)^k a )\;</math>
with ''k'' ranging from 0 to 2''n''-1; if the triangles are equilateral,
:<math>2a^2=\cos(\pi/n)-\cos(2\pi/n)\;</math>.

== 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 in a point|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 ''D₂'' as subgroups, and the octahedron, which has the larger rotation group '''O''' of order 24, which has four versions of ''D₃'' as subgroups.

== See also ==

*[[Prismatic uniform polyhedron]]

== External links ==

* {{MathWorld | urlname=Antiprism | title=Antiprism}}
* {{GlossaryForHyperspace | anchor=Antiprism | title=Antiprism}}
** {{GlossaryForHyperspace | anchor=Prismatic | title=Prismatic polytopes }}
*[http://home.comcast.net/~tpgettys/nonconvexprisms.html Nonconvex Prisms and Antiprisms]
*[http://www.software3d.com/Prisms.html Paper models of prisms and antiprisms]

[[Category:Polyhedra]]
[[Category:Uniform polyhedra]]
[[Category:Prismatoid polyhedra]]

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