# Hexagonal antiprism

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.

Uniform Hexagonal antiprism | |
---|---|

Type | Prismatic uniform polyhedron |

Elements | F = 14, E = 24V = 12 (χ = 2) |

Faces by sides | 12{3}+2{6} |

Schläfli symbol | s{2,12} sr{2,6} |

Wythoff symbol | | 2 2 6 |

Coxeter diagram | |

Symmetry group | D_{6d}, [2^{+},12], (2*6), order 24 |

Rotation group | D_{6}, [6,2]^{+}, (622), order 12 |

References | U_{77(d)} |

Dual | Hexagonal trapezohedron |

Properties | convex |

Vertex figure 3.3.3.6 |

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, 2*n* 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 d_{6d} symmetry, order 12.

## Related polyhedra

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

Uniform hexagonal dihedral spherical polyhedra | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [6,2], (*622) | [6,2]^{+}, (622) |
[6,2^{+}], (2*3) | ||||||||||||

{6,2} | t{6,2} | r{6,2} | t{2,6} | {2,6} | rr{6,2} | tr{6,2} | sr{6,2} | s{2,6} | ||||||

Duals to uniforms | ||||||||||||||

V6^{2} |
V12^{2} |
V6^{2} |
V4.4.6 | V2^{6} |
V4.4.6 | V4.4.12 | V3.3.3.6 | V3.3.3.3 |

Family of uniform antiprisms n.3.3.3 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Polyhedron | ||||||||||||

Tiling | ||||||||||||

Config. | V2.3.3.3 | 3.3.3.3 | 4.3.3.3 | 5.3.3.3 | 6.3.3.3 | 7.3.3.3 | 8.3.3.3 | 9.3.3.3 | 10.3.3.3 | 11.3.3.3 | 12.3.3.3 | ...∞.3.3.3 |

## External links

- Weisstein, Eric W. "Antiprism".
*MathWorld*. - Hexagonal Antiprism: Interactive Polyhedron model
- Virtual Reality Polyhedra www.georgehart.com: The Encyclopedia of Polyhedra
- VRML model
- Conway Notation for Polyhedra Try: "A6"