# Hexagonal antiprism

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

Rank | 3 |

Type | Uniform |

Notation | |

Bowers style acronym | Hap |

Coxeter diagram | s2s12o () |

Conway notation | A6 |

Elements | |

Faces | 12 triangles, 2 hexagons |

Edges | 12+12 |

Vertices | 12 |

Vertex figure | Isosceles trapezoid, edge lengths 1, 1, 1, √3 |

Measures (edge length 1) | |

Circumradius | |

Volume | |

Dihedral angles | 3–3: |

6–3: | |

Height | |

Central density | 1 |

Number of external pieces | 14 |

Level of complexity | 4 |

Related polytopes | |

Army | Hap |

Regiment | Hap |

Dual | Hexagonal antitegum |

Conjugate | Hexagonal retroprism |

Abstract & topological properties | |

Flag count | 96 |

Euler characteristic | 2 |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | (I_{2}(12)×A_{1})/2, order 24 |

Convex | Yes |

Nature | Tame |

The **hexagonal antiprism**, or **hap**, is a prismatic uniform polyhedron. It consists of 12 triangles and 2 hexagons. Each vertex joins one hexagon and three triangles. As the name suggests, it is an antiprism based on a hexagon.

## Vertex coordinates[edit | edit source]

A hexagonal antiprism of edge length 1 has vertex coordinates given by:

- ,
- ,
- ,
- .

## Representations[edit | edit source]

A hexagonal antiprism has the following Coxeter diagrams:

- s2s12o () (alternated dodecagonal prism)
- s2s6s () (alternated dihexagonal prism)
- xo6ox&#x (bases considered separately)

## General variant[edit | edit source]

The hexagonal antiprism has a general isogonal variant of the form xo6ox&#y that maintains its full symmetry. This variant uses isosceles triangles as sides.

If the base edges are of length b and the lacing edges are of length l, its height is given by .

The bases of the hexagonal antiprism are rotated from each other by an angle of 30°. If this angle is changed the result is more properly called a hexagonal gyroprism.

A notable case occurs as the alternation of the uniform dodecagonal prism. This specific case has base edges of length and side edges of length .

## Related polyhedra[edit | edit source]

A triangular cupola can be attached to a base of the hexagonal antiprism to form the gyroelongated triangular cupola. If a second triangular cupola is attached to the other base, the result is the gyroelongated triangular bicupola.

## External links[edit | edit source]

- Klitzing, Richard. "hap".
- Quickfur. "The Hexagonal Antiprism".

- Wikipedia contributors. "Hexagonal antiprism".
- McCooey, David. "Hexagonal Antiprism"