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Truncated order-5 square tiling | |
---|---|

Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic uniform tiling |

Vertex configuration | 8.8.5 |

Schläfli symbol | t{4,5} |

Wythoff symbol | 2 5 | 4 |

Coxeter diagram | |

Symmetry group | [5,4], (*542) |

Dual | Order-4 pentakis pentagonal tiling |

Properties | Vertex-transitive |

In geometry, the **truncated order-5 square tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{4,5}.

**Geometry** is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

In mathematics, **hyperbolic geometry** is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

In geometry, the **Schläfli symbol** is a notation of the form {p,q,r,...} that defines regular polytopes and tessellations.

Uniform pentagonal/square tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [5,4], (*542) | [5,4]^{+}, (542) | [5^{+},4], (5*2) | [5,4,1^{+}], (*552) | ||||||||

{5,4} | t{5,4} | r{5,4} | 2t{5,4}=t{4,5} | 2r{5,4}={4,5} | rr{5,4} | tr{5,4} | sr{5,4} | s{5,4} | h{4,5} | ||

Uniform duals | |||||||||||

V5^{4} | V4.10.10 | V4.5.4.5 | V5.8.8 | V4^{5} | V4.4.5.4 | V4.8.10 | V3.3.4.3.5 | V3.3.5.3.5 | V5^{5} |

*n42 symmetry mutation of truncated tilings: n.8.8 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry * n42 [n,4] | Spherical | Euclidean | Compact hyperbolic | Paracompact | |||||||

*242 [2,4] | *342 [3,4] | *442 [4,4] | *542 [5,4] | *642 [6,4] | *742 [7,4] | *842 [8,4]... | *∞42 [∞,4] | ||||

Truncated figures | |||||||||||

Config. | 2.8.8 | 3.8.8 | 4.8.8 | 5.8.8 | 6.8.8 | 7.8.8 | 8.8.8 | ∞.8.8 | |||

n-kis figures | |||||||||||

Config. | V2.8.8 | V3.8.8 | V4.8.8 | V5.8.8 | V6.8.8 | V7.8.8 | V8.8.8 | V∞.8.8 |

In geometry, the **snub tetrapentagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,4}.

In geometry, the **truncated tetrapentagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1,2}{4,5} or tr{4,5}.

In geometry, the **rhombitetrapentagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,2}{4,5}.

In geometry, the **truncated order-4 pentagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{5,4}.

In geometry, the **truncated order-5 pentagonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{5,5}, constructed from one pentagons and two decagons around every vertex.

In geometry, the **snub pentapentagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{5,5}, constructed from two regular pentagons and three equilateral triangles around every vertex.

In geometry, the **tetraheptagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{4,7}.

In geometry, the **truncated order-7 square tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{4,7}.

In geometry, the **order-7 square tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,7}.

In geometry, the **order-7 heptagonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {7,7}, constructed from seven heptagons around every vertex. As such, it is self-dual.

In geometry, the **truncated order-7 heptagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{7,7}, constructed from one heptagons and two tetrakaidecagons around every vertex.

In geometry, the **tritetratrigonal tiling** or **shieldotritetragonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{1,2}(4,3,3). It can also be named as a **cantic octagonal tiling**, h_{2}{8,3}.

In geometry, the **alternated order-4 hexagonal tiling** or **ditetragonal tritetratrigonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}.

In geometry, the **tritetratrigonal tiling** or **cantic order-4 hexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{(4,4,3)} or h_{2}{6,4}.

In geometry, the **truncated order-5 hexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{6,5}.

In geometry, the **pentahexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{6,5} or t_{1}{6,5}.

In geometry, the **rhombipentahexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,2}{6,5}.

In geometry, the **snub pentahexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,5}.

In geometry, the **snub triapeirotrigonal tiling** is a uniform tiling of the hyperbolic plane with a Schläfli symbol of s{3,∞}.

In geometry, the **order-8 pentagonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,8}.

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

**John Horton Conway** FRS is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He has also contributed to many branches of recreational mathematics, notably the invention of the cellular automaton called the Game of Life. Conway spent the first half of his long career at the University of Cambridge, in England, and the second half at Princeton University in New Jersey, where he now holds the title Professor Emeritus.

The **International Standard Book Number** (**ISBN**) is a numeric commercial book identifier which is intended to be unique. Publishers purchase ISBNs from an affiliate of the International ISBN Agency.

In hyperbolic geometry, a **uniform****hyperbolic tiling** is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent, and the tiling has a high degree of rotational and translational symmetry.

Wikimedia Commons has media related to . Uniform tiling 5-8-8 |

- Weisstein, Eric W. "Hyperbolic tiling".
*MathWorld*. - Weisstein, Eric W. "Poincaré hyperbolic disk".
*MathWorld*. - Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch

**Eric Wolfgang Weisstein** is an encyclopedist who created and maintains *MathWorld* and *Eric Weisstein's World of Science* (*ScienceWorld*). He is the author of the *CRC Concise Encyclopedia of Mathematics*. He currently works for Wolfram Research, Inc.

* MathWorld* is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana–Champaign.

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