# Hexagonal-cubic duoprism

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Hexagonal-cubic duoprism | |
---|---|

Rank | 5 |

Type | Uniform |

Notation | |

Bowers style acronym | Hacube |

Coxeter diagram | x6o x4o3o () |

Elements | |

Tera | 6 tesseracts, 6 square-hexagonal duoprisms |

Cells | 6+36 cubes, 12 hexagonal prisms |

Faces | 36+72 squares, 8 hexagons |

Edges | 48+72 |

Vertices | 48 |

Vertex figure | Triangular scalene, edge lengths √3 (top), √2 (base triangle and sides) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Diteral angles | Tes–cube–tes: 120° |

Tes–cube–shiddip: 90° | |

Shiddip–hip–shiddip: 90° | |

Height | 1 |

Central density | 1 |

Number of external pieces | 12 |

Level of complexity | 10 |

Related polytopes | |

Army | Hacube |

Regiment | Hacube |

Dual | Hexagonal-octahedral duotegum |

Conjugate | Hexagonal-cubic duoprism |

Abstract & topological properties | |

Flag count | 5760 |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | B_{3}×G_{2}, order 576 |

Convex | Yes |

Nature | Tame |

The **hexagonal-cubic duoprism** or **hacube**, also known as a **square-hexagonal duoprismatic prism**, is a convex uniform duoprism that consists of 6 tesseracts and 6 square-hexagonal duoprisms. Each vertex joins 2 tesseracts and 3 square-hexagonal duoprisms. It is a duoprism based on a square and a hexagonal prism, which makes it a convex segmentoteron.

This polyteron can be alternated into a triangular-tetrahedral duoantiprism, although it cannot be made uniform.

## Vertex coordinates[edit | edit source]

The vertices of a hexagonal-cubic duoprism of edge length 1 are given by:

## Representations[edit | edit source]

A hexagonal-cubic duoprism has the following Coxeter diagrams:

- x6o x4o3o (full symmetry)
- x3x x4o3o (hexagons as ditrigons)
- x x4o x6o (square-hexagonal duoprismatic prism)
- x x3x x4o
- x x x x6o (hexagonal prismatic prismatic prism)
- x x x x3x
- xx4oo xx6oo&#x (square-hexagonal duoprism atop square-hexagonal duoprism)
- xx xx xx6oo&#x
- xx4oo xx3xx&#x
- xx xx xx3xx&#x

## External links[edit | edit source]

- Klitzing, Richard. "hacube".