Truncated order-8 hexagonal tiling | |
---|---|

Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic uniform tiling |

Vertex configuration | 8.12.12 |

Schläfli symbol | t{6,8} |

Wythoff symbol | 6 |

Coxeter diagram | |

Symmetry group | [8,6], (*862) |

Dual | Order-6 octakis octagonal tiling |

Properties | Vertex-transitive |

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

This tiling can also be constructed from *664 symmetry, as t{(6,6,4)}.

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-6 octagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 7 forms with full [8,6] symmetry, and 7 with subsymmetry.

Uniform octagonal/hexagonal tilings | ||||||
---|---|---|---|---|---|---|

Symmetry: [8,6], (*862) | ||||||

{8,6} | t{8,6} | r{8,6} | 2t{8,6}=t{6,8} | 2r{8,6}={6,8} | rr{8,6} | tr{8,6} |

Uniform duals | ||||||

V8^{6} | V6.16.16 | V(6.8)^{2} | V8.12.12 | V6^{8} | V4.6.4.8 | V4.12.16 |

Alternations | ||||||

[1^{+},8,6](*466) | [8^{+},6](8*3) | [8,1^{+},6](*4232) | [8,6^{+}](6*4) | [8,6,1^{+}](*883) | [(8,6,2^{+})](2*43) | [8,6]^{+}(862) |

h{8,6} | s{8,6} | hr{8,6} | s{6,8} | h{6,8} | hrr{8,6} | sr{8,6} |

Alternation duals | ||||||

V(4.6)^{6} | V3.3.8.3.8.3 | V(3.4.4.4)^{2} | V3.4.3.4.3.6 | V(3.8)^{8} | V3.4^{5} | V3.3.6.3.8 |

The dual of the tiling represents the fundamental domains of (*664) orbifold symmetry. From [(6,6,4)] (*664) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to 862 symmetry by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1^{+},6,1^{+},6,1^{+},4)] (332332) is the commutator subgroup of [(6,6,4)].

A large subgroup is constructed [(6,6,4^{*})], index 8, as (4*33) with gyration points removed, becomes (*3^{8}), and another large subgroup is constructed [(6,6^{*},4)], index 12, as (6*32) with gyration points removed, becomes (*(32)^{6}).

Fundamental domains | | | | | ||
---|---|---|---|---|---|---|

Subgroup index | 1 | 2 | 4 | |||

Coxeter | [(6,6,4)] | [(1^{+},6,6,4)] | [(6,6,1^{+},4)] | [(6,1^{+},6,4)] | [(1^{+},6,6,1^{+},4)] | [(6^{+},6^{+},4)] |

Orbifold | *664 | *6362 | *4343 | 2*3333 | 332× | |

Coxeter | [(6,6^{+},4)] | [(6^{+},6,4)] | [(6,6,4^{+})] | [(6,1^{+},6,1^{+},4)] | [(1^{+},6,1^{+},6,4)] | |

Orbifold | 6*32 | 4*33 | 3*3232 | |||

Direct subgroups | ||||||

Subgroup index | 2 | 4 | 8 | |||

Coxeter | [(6,6,4)]^{+} | [(1^{+},6,6^{+},4)] | [(6^{+},6,1^{+},4)] | [(6,1^{+},6,4^{+})] | [(6^{+},6^{+},4^{+})] = [(1^{+},6,1^{+},6,1^{+},4)]= | |

Orbifold | 664 | 6362 | 4343 | 332332 |

Wikimedia Commons has media related to Uniform tiling 8-12-12 . |

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 **truncated tetrahexagonal tiling** is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{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 order-6 square tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,6}.

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 **order-8 triangular tiling** is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of *{3,8}*, having eight regular triangles around each vertex.

In geometry, the **truncated order-6 hexagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{6,6}. It can also be identically constructed as a **cantic order-6 square tiling**, h_{2}{4,6}

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 **truncated order-4 octagonal tiling** is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t_{0,1}{8,4}. A secondary construction t_{0,1,2}{8,8} is called a **truncated octaoctagonal tiling** with two colors of hexakaidecagons.

In geometry, the **truncated tetraoctagonal tiling** is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.

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-3 apeirogonal tiling** is a regular tiling of the hyperbolic plane. It is represented by the Schläfli symbol {∞,3}, having three regular apeirogons around each vertex. Each apeirogon is inscribed in a horocycle.

In geometry, the **truncated triapeirogonal tiling** is a uniform tiling of the hyperbolic plane with a Schläfli symbol of tr{∞,3}.

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

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

In geometry, the **truncated tetraapeirogonal tiling** is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

In geometry, the **truncated hexaoctagonal tiling** is a semiregular tiling of the hyperbolic plane. There are one square, one dodecagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,6}.

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

In geometry, the **hexaoctagonal tiling** is a uniform tiling of the hyperbolic plane.

- 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.

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