Infinite-order hexagonal tiling | |
---|---|

Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic regular tiling |

Vertex configuration | 6^{∞} |

Schläfli symbol | {6,∞} |

Wythoff symbol | ∞ | 6 2 |

Coxeter diagram | |

Symmetry group | [∞,6], (*∞62) |

Dual | Order-6 apeirogonal tiling |

Properties | Vertex-transitive, edge-transitive, face-transitive |

In 2-dimensional hyperbolic geometry, the **infinite-order hexagonal tiling** is a regular tiling. It has Schläfli symbol of {6,∞}. All vertices are *ideal*, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

## Symmetry

There is a half symmetry form, , seen with alternating colors:

## Related polyhedra and tiling

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (6^{n}).

*n62 symmetry mutation of regular tilings: {6,n}
| ||||||||
---|---|---|---|---|---|---|---|---|

Spherical | Euclidean | Hyperbolic tilings | ||||||

{6,2} |
{6,3} |
{6,4} |
{6,5} |
{6,6} |
{6,7} |
{6,8} |
... | {6,∞} |

## See also

Wikimedia Commons has media related to .Infinite-order pentagonal tiling |

## References

- John H. Conway; Heidi Burgiel; Chaim Goodman-Strass (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations".
*The Symmetries of Things*. ISBN 978-1-56881-220-5. - H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space".
*The Beauty of Geometry: Twelve Essays*. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.