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Truncated pentagonal tiling | |
---|---|

Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic uniform tiling |

Vertex configuration | 4.10.10 |

Schläfli symbol | t{5,4} |

Wythoff symbol | 2 4 | 5 2 5 5 | |

Coxeter diagram | |

Symmetry group | [5,4], (*542) [5,5], (*552) |

Dual | Order-5 tetrakis square tiling |

Properties | Vertex-transitive |

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}.

**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.

A half symmetry [1+,4,5] = [5,5] coloring can be constructed with two colors of decagons. This coloring is called a *truncated pentapentagonal tiling*.

There is only one subgroup of [5,5], [5,5]^{+}, removing all the mirrors. This symmetry can be doubled to 542 symmetry by adding a bisecting mirror.

Type | Reflective domains | Rotational symmetry |
---|---|---|

Index | 1 | 2 |

Diagram | ||

Coxeter (orbifold) | [5,5] = (*552) | [5,5]^{+} = (552) |

*n42 symmetry mutation of truncated tilings: 4.2n.2n | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

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

*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. | 4.4.4 | 4.6.6 | 4.8.8 | 4.10.10 | 4.12.12 | 4.14.14 | 4.16.16 | 4.∞.∞ | |||

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

Config. | V4.4.4 | V4.6.6 | V4.8.8 | V4.10.10 | V4.12.12 | V4.14.14 | V4.16.16 | V4.∞.∞ |

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} |

Uniform pentapentagonal tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [5,5], (*552) | [5,5]^{+}, (552) | ||||||||||

= | = | = | = | = | = | = | = | ||||

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

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

V5.5.5.5.5 | V5.10.10 | V5.5.5.5 | V5.10.10 | V5.5.5.5.5 | V4.5.4.5 | V4.10.10 | V3.3.5.3.5 |

In geometry, the **truncated triheptagonal tiling** is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of *tr*{7,3}.

In geometry, the **order-4 pentagonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,4}. It can also be called a **pentapentagonal tiling** in a bicolored quasiregular form.

In geometry, the **octagonal tiling** is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of *{8,3}*, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

In geometry, the **truncated trioctagonal tiling** is a semiregular tiling of the hyperbolic plane. There are one square, one hexagon, and one hexadecagon (16-sides) on each vertex. It has Schläfli symbol of *tr*{8,3}.

In geometry, the **order-4 hexagonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

In geometry, the **rhombitetrahexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

In geometry, the **truncated order-4 hexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,4}. A secondary construction tr{6,6} is called a **truncated hexahexagonal tiling** with two colors of dodecagons.

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

In geometry, the **truncated order-8 triangular tiling** is a semiregular tiling of the hyperbolic plane. There are two hexagons and one octagon on each vertex. It has Schläfli symbol of t{3,8}.

In geometry, the **rhombitrioctagonal tiling** is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

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

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

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

In geometry, the **rhombitetraoctagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.

In geometry, the **truncated order-8 octagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{8,8}.

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

In geometry, the **truncated order-6 pentagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{1,2}{6,5}.

In geometry, the **truncated order-6 octagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{8,6}.

- 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.

Wikimedia Commons has media related to . Uniform tiling 4-10-10 |

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.

- 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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