# Order-∞ triangular tiling

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Order-∞ triangular tiling | |
---|---|

Rank | 3 |

Type | Regular, paracompact |

Space | Hyperbolic |

Notation | |

Bowers style acronym | Aztrat |

Coxeter diagram | o∞o3x () |

Schläfli symbol | {3,∞} |

Elements | |

Faces | 2NM Triangles |

Edges | 3NM |

Vertices | 6N |

Vertex figure | Apeirogon, edge length 1 |

Measures (edge length 1) | |

Circumradius | |

Related polytopes | |

Army | Aztrat |

Regiment | Aztrat |

Dual | Order-3 apeirogonal tiling |

Topological properties | |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | [∞,3] |

Convex | Yes |

The **order-∞ triangular tiling** or **trazat**, also called the **infinite-order triangular tiling** is a paracompact regular tiling of the hyperbolic plane. Infinitely many ideal triangles join at each vertex. All vertices are ideal points at infinity.

## Representations[edit | edit source]

An order–∞ triangular tiling has the following Coxeter diagrams:

- o∞o3x (full symmetry)
- o3x3o∞*a (triangles of two types)

## Related polytopes[edit | edit source]

Name | OBSA | Schläfli symbol | CD diagram | Picture |
---|---|---|---|---|

Apeirogonal tiling | azat | {∞,3} | ||

Truncated apeirogonal tiling | tazat | t{∞,3} | ||

Triapeirogonal tiling | tazt | r{∞,3} | ||

Truncated order-∞ triangular tiling | taztrat | t{3,∞} | ||

Order-∞ triangular tiling | aztrat | {3,∞} | ||

Small rhombitriapeirogonal tiling | srotazt | rr{∞,3} | ||

Great rhombitriapeirogonal tiling | grotazt | tr{∞,3} | ||

Snub triapeirogonal tiling | sr{∞,3} |

## External links[edit | edit source]

- Klitzing, Richard. "Aztrat".

- Wikipedia Contributors. "Infinite-order triangular tiling".