Truncated order-7 triangular tiling | |
---|---|

Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic uniform tiling |

Vertex configuration | 7.6.6 |

Schläfli symbol | t{3,7} |

Wythoff symbol | 3 |

Coxeter diagram | |

Symmetry group | [7,3], (*732) |

Dual | Heptakis heptagonal tiling |

Properties | Vertex-transitive |

In geometry, the **order-7 truncated triangular tiling**, sometimes called the **hyperbolic soccerball**,^{ [1] } is a semiregular tiling of the hyperbolic plane. There are two hexagons and one heptagon on each vertex, forming a pattern similar to a conventional soccer ball (truncated icosahedron) with heptagons in place of pentagons. It has Schläfli symbol of t{3,7}.

This tiling is called a **hyperbolic soccerball** (football) for its similarity to the truncated icosahedron pattern used on soccer balls. Small portions of it as a hyperbolic surface can be constructed in 3-space.

A truncated icosahedron as a polyhedron and a ball | The Euclidean hexagonal tiling colored as truncated triangular tiling | A paper construction of a hyperbolic soccerball |

The dual tiling is called a *heptakis heptagonal tiling*, named for being constructible as a heptagonal tiling with every heptagon divided into seven triangles by the center point.

This hyperbolic tiling is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (n.6.6), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated tilings: n.6.6 | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Sym. * n42[n,3] | Spherical | Euclid. | Compact | Parac. | Noncompact hyperbolic | |||||||

*232 [2,3] | *332 [3,3] | *432 [4,3] | *532 [5,3] | *632 [6,3] | *732 [7,3] | *832 [8,3]... | *∞32 [∞,3] | [12i,3] | [9i,3] | [6i,3] | ||

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

Config. | 2.6.6 | 3.6.6 | 4.6.6 | 5.6.6 | 6.6.6 | 7.6.6 | 8.6.6 | ∞.6.6 | 12i.6.6 | 9i.6.6 | 6i.6.6 | |

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

Config. | V2.6.6 | V3.6.6 | V4.6.6 | V5.6.6 | V6.6.6 | V7.6.6 | V8.6.6 | V∞.6.6 | V12i.6.6 | V9i.6.6 | V6i.6.6 |

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform heptagonal/triangular tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [7,3], (*732) | [7,3]^{+}, (732) | ||||||||||

{7,3} | t{7,3} | r{7,3} | t{3,7} | {3,7} | rr{7,3} | tr{7,3} | sr{7,3} | ||||

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

V7^{3} | V3.14.14 | V3.7.3.7 | V6.6.7 | V3^{7} | V3.4.7.4 | V4.6.14 | V3.3.3.3.7 |

This tiling features prominently in HyperRogue.

Wikimedia Commons has media related to . Uniform tiling 6-6-7 |

In geometry, the **heptagonal tiling** is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons around each vertex.

In geometry, the **order-7 triangular tiling** is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.

In geometry, the **triheptagonal tiling** is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.

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 **truncated heptagonal tiling** is a semiregular tiling of the hyperbolic plane. There are one triangle and two tetradecagons on each vertex. It has Schläfli symbol of *t*{7,3}. The tiling has a vertex configuration of 3.14.14.

In geometry, the **rhombitriheptagonal tiling** is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.

In geometry, the **order-3 snub heptagonal tiling** is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex. It has Schläfli symbol of *sr{7,3}*. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol *sr{7,4}*.

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

In geometry, the **order-3 snub octagonal tiling** is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of *sr{8,3}*.

In geometry, the **Truncated octagonal tiling** is a semiregular tiling of the hyperbolic plane. There is one triangle and two hexakaidecagons on each vertex. It has Schläfli symbol of *t*{8,3}.

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 **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 **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 **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 **snub heptaheptagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{7,7}, constructed from two regular heptagons and three equilateral triangles around every vertex.

- 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.

- 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
- Geometric explorations on the hyperbolic football by Frank Sottile

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Images, videos and audio are available under their respective licenses.