# Hexagonal tiling honeycomb

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Hexagonal tiling honeycomb | |
---|---|

Rank | 4 |

Type | Regular, paracompact |

Space | Hyperbolic |

Notation | |

Bowers style acronym | Hexah |

Coxeter diagram | x6o3o3o () |

Schläfli symbol | {6,3,3} |

Elements | |

Cells | 2N hexagonal tilings |

Faces | NM hexagons |

Edges | 2NM |

Vertices | NM |

Vertex figure | Tetrahedron, edge length √3 |

Measures (edge length 1) | |

Circumradius | |

Related polytopes | |

Army | Hexah |

Regiment | Hexah |

Dual | Tetrahedral honeycomb |

Petrie dual | Petrial hexagonal tiling honeycomb |

Abstract & topological properties | |

Orientable | Yes |

Properties | |

Symmetry | [6,3,3] |

Convex | Yes |

The **hexagonal tiling honeycomb**, also known as the **order-3 hexagonal tiling honeycomb**, is a paracompact regular tiling of 3D hyperbolic space. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity. 3 hexagonal tilings meet at each edge, and 4 meet at each vertex.

It can be seen as a truncated triangular tiling honeycomb or bitruncated order-6 hexagonal tiling honeycomb.

This honeycomb can be alternated into an alternated hexagonal tiling honeycomb, which is uniform.

## Representations[edit | edit source]

The hexagonal tiling honeycomb has the following Coxeter diagrams:

- x6o3o3o () (full symmetry)
- x3x6o3o () (as truncated triangular tiling honeycomb)
- o6x3x6o () (as bitruncated order-6 hexagonal tiling honeycomb)
- o6x3x3x3*b () (half symmetry of bitruncated order-6 hexagonal tiling honeycomb)
- x3x3x3x3*a3*c *b3*d () (quarter symmetry of bitruncated order-6 hexagonal tiling honeycomb)

## Related polytopes[edit | edit source]

## External links[edit | edit source]

- Klitzing, Richard. "hexah".

- Wikipedia Contributors. "Hexagonal tiling honeycomb".
- lllllllllwith10ls. "Category 1: Regulars" (#5).