# Hexagonal-great rhombicuboctahedral duoprism

Hexagonal-great rhombicuboctahedral duoprism | |
---|---|

Rank | 5 |

Type | Uniform |

Notation | |

Bowers style acronym | Hagirco |

Coxeter diagram | x6o x4x3x |

Elements | |

Tera | 12 square-hexagonal duoprisms, 8 hexagonal duoprisms, 6 hexagonal-octagonal duoprisms, 6 great rhombicuboctahedral prisms |

Cells | 96 cubes, 24+24+24+48 hexagonal prisms, 36 octagonal prisms, 6 great rhombicuboctahedra |

Faces | 72+144+144+144 squares, 48+48 hexagons, 36 octagons |

Edges | 144+144+144+288 |

Vertices | 288 |

Vertex figure | Mirror-symmetric pentachoron, edge lengths √2, √3, √2+√2 (base triangle), √3 (top edge), √2 (side edges) |

Measures (edge length 1) | |

Circumradius | |

Hypervolume | |

Diteral angles | Shiddip–hip–hiddip: |

Shiddip–hip–hodip: 135° | |

Hiddip–hip–hodip: | |

Gircope–girco–gircope: 120° | |

Shiddip–cube–gircope: 90° | |

Hiddip–hip–gircope: 90° | |

Hodip–op–gircope: 90° | |

Central density | 1 |

Number of external pieces | 32 |

Level of complexity | 60 |

Related polytopes | |

Army | Hagirco |

Regiment | Hagirco |

Dual | Hexagonal-disdyakis dodecahedral duotegum |

Conjugate | Hexagonal-quasitruncated cuboctahedral duoprism |

Abstract & topological properties | |

Euler characteristic | 2 |

Orientable | Yes |

Properties | |

Symmetry | B_{3}×G_{2}, order 576 |

Convex | Yes |

Nature | Tame |

The **hexagonal-great rhombicuboctahedral duoprism** or **hagirco** is a convex uniform duoprism that consists of 6 great rhombicuboctahedral prisms, 6 hexagonal-octagonal duoprisms, 8 hexagonal duoprisms, and 12 square-hexagonal duoprisms. Each vertex joins 2 great rhombicuboctahedral prisms, 1 square-hexagonal duoprism, 1 hexagonal duoprism, and 1 hexagonal-octagonal duoprism.

This polyteron can be alternated into a triangular-snub cubic duoantiprism, although it cannot be made uniform. The great rhombicuboctahedra can also be edge-snubbed to create a triangular-pyritohedral prismantiprismoid, which is also nonuniform.

## Vertex coordinates[edit | edit source]

The vertices of a hexagonal-great rhombicuboctahedral duoprism of edge length 1 are given by all permutations of the last three coordinates of:

## Representations[edit | edit source]

A hexagonal-great rhombicuboctahedral duoprism has the following Coxeter diagrams:

- x6o x4x3x (full symmetry)
- x3x x4x3x (hexagons as ditrigons)

## External links[edit | edit source]

- Klitzing, Richard. "hagirco".