# Great rhombi-tetrahedral-octahedral honeycomb

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Great rhombi-tetrahedral-octahedral honeycomb | |
---|---|

Rank | 4 |

Type | uniform |

Space | Euclidean |

Notation | |

Bowers style acronym | Gratoh |

Coxeter diagram | x3x3o *b4x () |

Elements | |

Cells | 2N truncated tetrahedra, N truncated cubes, N great rhombicuboctahedra |

Faces | 8N triangles, 6N squares, 8N hexagons, 6N octagons |

Edges | 12N+12N+24N |

Vertices | 24N |

Vertex figure | Sphenoid, edge lengths 1 (1), √2 (1), √3 (2), and √2+√2 (2) |

Measures (edge length 1) | |

Vertex density | |

Dual cell volume | |

Related polytopes | |

Army | Gratoh |

Regiment | Gratoh |

Dual | Half sphenoidal honeycomb |

Conjugate | Great quasirhombi-tetrahedral-octahedral honeycomb |

Abstract & topological properties | |

Orientable | Yes |

Properties | |

Symmetry | S_{4} |

Convex | Yes |

The **great rhombi-tetrahedral-octahedral honeycomb**, or **gratoh**, also known as the **runcicantic cubic honeycomb**, is a convex uniform honeycomb. 1 truncated tetrahedron, 1 truncated cube, and 2 great rhombicuboctahedra join at each vertex of this honeycomb. It can be formed as an alternated faceting from the prismatorhombated cubic honeycomb, or alternately as the bitruncation of the tetrahedral-octahedral honeycomb.

## Representations[edit | edit source]

A great rhombated tetrahedral-octahedral honeycomb has the following Coxeter diagrams:

- x3x3o *b4x (full symmetry)
- s4o3x4x (as alternated faceting)

## Gallery[edit | edit source]

## External links[edit | edit source]

- Klitzing, Richard. "gratoh".

- Wikipedia Contributors. "Runcicantic cubic honeycomb".
- Binnendyk, Eric. "Category 5: Greater Truncates" (#93).