# Hexagonal-octagonal duoprismatic prism

Hexagonal-octagonal duoprismatic prism | |
---|---|

Rank | 5 |

Type | Uniform |

Notation | |

Bowers style acronym | Haop |

Coxeter diagram | x x6o x8o |

Elements | |

Tera | 8 square-hexagonal duoprisms, 6 square-octagonal duoprisms, 2 hexagonal-octagonal duoprisms |

Cells | 48 cubes, 6+12 octagonal prisms, 8+16 hexagonal prisms |

Faces | 48+48+96 squares, 16 hexagons, 12 octagons |

Edges | 48+96+96 |

Vertices | 96 |

Vertex figure | Digonal disphenoidal pyramid, edge lengths √3 (disphenoid base 1), √2+√2 (disphenoid base 2), √2 (remaining edges) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Diteral angles | Shiddip–hip–shiddip: 135° |

Sodip–op–sodip: 120° | |

Sodip–cube–shiddip: 90° | |

Hodip–hip–shiddip: 90° | |

Sodip–op–hodip: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 16 |

Level of complexity | 30 |

Related polytopes | |

Army | Haop |

Regiment | Haop |

Dual | Hexagonal-octagonal duotegmatic tegum |

Conjugate | Hexagonal-octagrammic duoprismatic prism |

Abstract & topological properties | |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | G_{2}×I_{2}(8)×A_{1}, order 384 |

Convex | Yes |

Nature | Tame |

The **hexagonal-octagonal duoprismatic prism** or **haop**, also known as the **hexagonal-octagonal prismatic duoprism**, is a convex uniform duoprism that consists of 2 hexagonal-octagonal duoprisms, 6 square-octagonal duoprisms, and 8 square-hexagonal duoprisms. Each vertex joins 2 square-hexagonal duoprisms, 2 square-octagonal duoprisms, and 1 hexagonal-octagonal duoprism. Being a prism based on an orbiform polytope, it is also a convex segmentoteron.

This polyteron can be alternated into a triangular-square duoantiprismatic antiprism, although it cannot be made uniform. The octagons can also be alternated into long rectangles to create a triangular-square prismatic prismantiprismoid, which is also nonuniform.

## Vertex coordinates[edit | edit source]

The vertices of a hexagonal-octagonal duoprismatic prism of edge length 1 are given by all permutations of the third and fourth coordinates of:

## Representations[edit | edit source]

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

- x x6o x8o (full symmetry)
- x x3x x8o (hexagons as ditrigons)
- x x6o x4x (octagons as ditetragons)
- x x3x x4x
- xx6oo xx8oo&#x (hexagonal-octagonal duoprism atop hexagonal-octagonal duoprism)
- xx3xx xx8oo&#x
- xx6oo xx4xx&#x
- xx3xx xx4xx&#x