# Order-∞ apeirogonal tiling

The **order-∞ apeirogonal tiling** or **infinite-order apeirogonal tiling** is a paracompact regular tiling of the hyperbolic plane. Infinitely many apeirogons join at each vertex. It is self-dual and abstractly self-Petrial.

Order-∞ apeirogonal tiling | |
---|---|

Rank | 3 |

Dimension | 2 |

Type | Regular, paracompact |

Space | Hyperbolic |

Notation | |

Bowers style acronym | Azazat |

Coxeter diagram | x∞o∞o () |

Schläfli symbol | {∞,∞} |

Elements | |

Faces | 2N apeirogons |

Edges | NM |

Vertices | 2N |

Vertex figure | Apeirogon, edge length 2 |

Petrie polygons | 2N zigzags |

Holes | N apeirogons |

Measures (edge length 1) | |

Circumradius | 0 |

Related polytopes | |

Army | Azazat |

Regiment | Azazat |

Dual | Order-∞ apeirogonal tiling |

Petrie dual | Petrial order-∞ apeirogonal tiling |

φ 2 | Order-∞ apeirogonal tiling |

Abstract & topological properties | |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | [∞,∞] |

Convex | Yes |

## Representations Edit

The order-∞ apeirogonal tiling has the following Coxeter diagrams:

- x∞o∞o ( ) (full symmetry)
- x∞o∞o∞*a ( ) (apeirogons of two types)

## External links Edit

- Klitzing, Richard. "azazat".
- Wikipedia contributors. "Infinite-order apeirogonal tiling".