# Hexagonal antiprism

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 |

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, 2*n* isosceles triangles.

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

## 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 | ||||||||||||||
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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

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