# Order-∞ square tiling

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Order-∞ square tiling | |
---|---|

Rank | 3 |

Type | Regular, paracompact |

Space | Hyperbolic |

Notation | |

Bowers style acronym | Asquat |

Coxeter diagram | o∞o4x () |

Schläfli symbol | {4,∞} |

Elements | |

Faces | NM squares |

Edges | 2NM |

Vertices | 4N |

Vertex figure | Apeirogon, edge length √2 |

Measures (edge length 1) | |

Circumradius | |

Related polytopes | |

Army | Asquat |

Regiment | Asquat |

Dual | Order-4 apeirogonal tiling |

Topological properties | |

Surface | Sphere |

Orientable | Yes |

Genus | 0 |

Properties | |

Symmetry | [∞,4] |

Convex | Yes |

The **order-∞ square tiling** or **infinite-order square tiling** is a paracompact regular tiling of the hyperbolic plane. Infinitely many ideal squares join at each vertex. All vertices are ideal points at infinity.

The tiling can be alternated to produce the order-∞ apeirogonal tiling.

## Representations[edit | edit source]

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

- o∞o4x (full symmetry)
- o4x4o∞*a (squares of two types)

## Related polytopes[edit | edit source]

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

Order-4 apeirogonal tiling | squazat | {∞,4} | ||

Truncated order-4 apeirogonal tiling | tosquazat | t{∞,4} | ||

Tetraapeirogonal tiling | tezt | r{∞,4} | ||

Truncated order-∞ square tiling | tazsquat | t{4,∞} | ||

Order-∞ square tiling | azsquat | {4,∞} | ||

Small rhombitetraapeirogonal tiling | srotezt | rr{∞,4} | ||

Great rhombitetraapeirogonal tiling | grotezt | tr{∞,4} | ||

Snub tetraapeirogonal tiling | sr{∞,4} |

## External links[edit | edit source]

- Klitzing, Richard. "asquat".

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