Hexagonal Antiprism
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Hexagonal Antiprism
Uniform hexagonal antiprism
Hexagonal antiprism.png
Type Prismatic uniform polyhedron
Elements F = 14, E = 24
V = 12 (? = 2)
Faces by sides 12{3}+2{6}
Schläfli symbol s{2,12}
sr{2,6}
Wythoff symbol | 2 2 6
Coxeter diagram CDel node h.pngCDel 2.pngCDel node h.pngCDel 12.pngCDel node.png
CDel node h.pngCDel 2.pngCDel node h.pngCDel 6.pngCDel node h.png
Symmetry group D6d, [2+,12], (2*6), order 24
Rotation group D6, [6,2]+, (622), order 12
References U77(d)
Dual Hexagonal trapezohedron
Properties convex
Hexagonal antiprism vertfig.png
Vertex figure
3.3.3.6

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 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 6-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.

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 triangles. Its vertex configuration is 3.3/2.3.6, with one triangle retrograde. It has d6d symmetry, order 12.

Crossed hexagonal antiprism.png

Related polyhedra

The hexagonal faces can be replaced by coplanar triangles, leading to a nonconvex polyhedron with 24 equilateral triangles.

Augmented hexagonal antiprism flat.png

External links

  • Weisstein, Eric W. "Antiprism". MathWorld.
  • Hexagonal Antiprism: Interactive Polyhedron model
  • Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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