# Hexagonal-heptagonal duoprismatic prism

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Hexagonal-heptagonal duoprismatic prism | |
---|---|

Rank | 5 |

Type | Uniform |

Notation | |

Bowers style acronym | Hahep |

Coxeter diagram | x x6o x7o |

Elements | |

Tera | 7 square-hexagonal duoprisms, 6 square-heptagonal duoprisms, 2 hexagonal-heptagonal duoprisms |

Cells | 42 cubes, 6+12 heptagonal prisms, 7+14 hexagonal prisms |

Faces | 42+42+84 squares, 14 hexagons, 12 heptagons |

Edges | 42+84+84 |

Vertices | 84 |

Vertex figure | Digonal disphenoidal pyramid, edge lengths √3 (disphenoid base 1), 2cos(π/7) (disphenoid base 2), √2 (remaining edges) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Diteral angles | Shiddip–hip–shiddip: |

Squahedip–hep–squahedip: 120° | |

Squahedip–cube–shiddip: 90° | |

Haheddip–hip–shiddip: 90° | |

Squahedip–hep–haheddip: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 15 |

Level of complexity | 30 |

Related polytopes | |

Army | Hahep |

Regiment | Hahep |

Dual | Hexagonal-heptagonal duotegmatic tegum |

Conjugates | Hexagonal-heptagrammic duoprismatic prism, Hexagonal-great heptagrammic duoprismatic prism |

Abstract & topological properties | |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | G_{2}×I_{2}(7)×A_{1}, order 336 |

Convex | Yes |

Nature | Tame |

The **hexagonal-heptagonal duoprismatic prism** or **hahep**, also known as the **hexagonal-heptagonal prismatic duoprism**, is a convex uniform duoprism that consists of 2 hexagonal-heptagonal duoprisms, 6 square-heptagonal duoprisms, and 7 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-heptagonal duoprisms, and 1 hexagonal-heptagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

## Vertex coordinates[edit | edit source]

The vertices of a hexagonal-heptagonal duoprismatic prism of edge length 2sin(π/7) are given by:

where j = 2, 4, 6.

## Representations[edit | edit source]

A hexagonal-heptagonal duoprismatic prism has the following Coxeter diagrams:

- x x6o x7o (full symmetry)
- x x3x x7o (hexagons as ditrigons)
- xx6oo xx7oo&#x (hexagonal-heptagonal duoprism atop hexagonal-heptagonal duoprism)
- xx3xx xx7oo&#x