View source for Hexagonal antiprism

{{short description|Antiprism with 6-sided caps}}
{{Prism polyhedra db|Prism polyhedron stat table|AP6}}

In [[geometry]], the '''hexagonal antiprism''' is the 4th in an infinite set of [[antiprisms]] formed by an even-numbered sequence of triangle sides closed by two [[polygon]] caps.

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 [[quadrilateral]]s.

In the case of a regular ''n''-sided base, one usually considers the case where its copy is twisted by an angle {{math|{{sfrac|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 {{nowrap|[[n-gon|{{mvar|n}}-gonal]]}} bases and, connecting those bases, {{math|2''n''}} [[isosceles triangle]]s.

If faces are all regular, it is a [[semiregular polyhedron]].

== Crossed antiprism==
A '''crossed hexagonal antiprism''' is a [[star polyhedron]], topologically identical to the convex ''hexagonal antiprism'' with the same [[vertex arrangement]], but it can't be made uniform; the sides are [[isosceles triangle]]s. Its [[vertex configuration]] is 3.3/2.3.6, with one triangle retrograde. It has D<sub>6d</sub> symmetry, order 24.
:[[File:Crossed hexagonal antiprism.png|240px]]

== Related polyhedra==
The hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles.
:[[File:Augmented hexagonal antiprism flat.png|100px]]

{{Hexagonal dihedral truncations}}

{{UniformAntiprisms}}

== External links ==
* {{mathworld | urlname = Antiprism | title = Antiprism }}
* [https://web.archive.org/web/20080616071808/http://polyhedra.org/poly/show/29/hexagonal_antiprism Hexagonal Antiprism: Interactive Polyhedron model]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
** [[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrl model] {{Webarchive|url=https://web.archive.org/web/20070207112806/http://www.georgehart.com/virtual-polyhedra/vrml/hexagonal_antiprism.wrl |date=2007-02-07 }}
* [http://levskaya.github.io/polyhedronisme/?recipe=C100A6 polyhedronisme] A6

[[Category:Prismatoid polyhedra]]
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