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Truncated tetraoctagonal tiling | |
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Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic uniform tiling |

Vertex configuration | 4.8.16 |

Schläfli symbol | tr{8,4} or |

Wythoff symbol | 2 8 4 | |

Coxeter diagram | |

Symmetry group | [8,4], (*842) |

Dual | Order-4-8 kisrhombille tiling |

Properties | Vertex-transitive |

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}.

The dual tiling is called an order-4-8 kisrhombille tiling, made as a complete bisection of the order-4 octagonal tiling, here with triangles are shown with alternating colors. This tiling represents the fundamental triangular domains of [8,4] (*842) symmetry. |

There are 15 subgroups constructed from [8,4] by mirror removal and alternation. 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 subgroup index-8 group, [1^{+},8,1^{+},4,1^{+}] (4242) is the commutator subgroup of [8,4].

A larger subgroup is constructed as [8,4*], index 8, as [8,4^{+}], (4*4) with gyration points removed, becomes (*4444) or (*4^{4}), and another [8*,4], index 16 as [8^{+},4], (8*2) with gyration points removed as (*22222222) or (*2^{8}). And their direct subgroups [8,4*]^{+}, [8*,4]^{+}, subgroup indices 16 and 32 respectively, can be given in orbifold notation as (4444) and (22222222).

Small index subgroups of [8,4] (*842) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Index | 1 | 2 | 4 | ||||||||

Diagram | |||||||||||

Coxeter | [8,4] | [1^{+},8,4] | [8,4,1^{+}] | [8,1^{+},4] | [1^{+},8,4,1^{+}] | [8^{+},4^{+}] | |||||

Orbifold | *842 | *444 | *882 | *4222 | *4242 | 42× | |||||

Semidirect subgroups | |||||||||||

Diagram | |||||||||||

Coxeter | [8,4^{+}] | [8^{+},4] | [(8,4,2^{+})] | [8,1^{+},4,1^{+}]= | [1^{+},8,1^{+},4]= | ||||||

Orbifold | 4*4 | 8*2 | 2*42 | 2*44 | 4*22 | ||||||

Direct subgroups | |||||||||||

Index | 2 | 4 | 8 | ||||||||

Diagram | |||||||||||

Coxeter | [8,4]^{+} | [8,4^{+}]^{+} | [8^{+},4]^{+} | [8,1^{+},4]^{+} | [8^{+},4^{+}]^{+} = [1^{+},8,1^{+},4,1^{+}] | ||||||

Orbifold | 842 | 444 | 882 | 4222 | 4242 | ||||||

Radical subgroups | |||||||||||

Index | 8 | 16 | 32 | ||||||||

Diagram | |||||||||||

Coxeter | [8,4*] | [8*,4] | [8,4*]^{+} | [8*,4]^{+} | |||||||

Orbifold | *4444 | *22222222 | 4444 | 22222222 |

From a Wythoff construction there are fourteen hyperbolic uniform tilings that can be based from the regular order-4 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 [8,4] symmetry, and 7 with subsymmetry.

Uniform octagonal/square tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

[8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) | |||||||||||

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

{8,4} | t{8,4} | r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} | |||||

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

V8^{4} | V4.16.16 | V(4.8)^{2} | V8.8.8 | V4^{8} | V4.4.4.8 | V4.8.16 | |||||

Alternations | |||||||||||

[1^{+},8,4](*444) | [8^{+},4](8*2) | [8,1^{+},4](*4222) | [8,4^{+}](4*4) | [8,4,1^{+}](*882) | [(8,4,2^{+})](2*42) | [8,4]^{+}(842) | |||||

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

h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} | |||||

Alternation duals | |||||||||||

V(4.4)^{4} | V3.(3.8)^{2} | V(4.4.4)^{2} | V(3.4)^{3} | V8^{8} | V4.4^{4} | V3.3.4.3.8 |

*n42 symmetry mutation of omnitruncated tilings: 4.8.2n | ||||||||
---|---|---|---|---|---|---|---|---|

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] | |

Omnitruncated figure | 4.8.4 | 4.8.6 | 4.8.8 | 4.8.10 | 4.8.12 | 4.8.14 | 4.8.16 | 4.8.∞ |

Omnitruncated duals | V4.8.4 | V4.8.6 | V4.8.8 | V4.8.10 | V4.8.12 | V4.8.14 | V4.8.16 | V4.8.∞ |

*nn2 symmetry mutations of omnitruncated tilings: 4.2n.2n | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Symmetry * nn2[n,n] | Spherical | Euclidean | Compact hyperbolic | Paracomp. | ||||||||||

*222 [2,2] | *332 [3,3] | *442 [4,4] | *552 [5,5] | *662 [6,6] | *772 [7,7] | *882 [8,8]... | *∞∞2 [∞,∞] | |||||||

Figure | ||||||||||||||

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

Dual | ||||||||||||||

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

Wikimedia Commons has media related to . Uniform tiling 4-8-16 |

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 **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-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 **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 **order-4 octagonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a **octaoctagonal tiling**, and Schläfli symbol of r{8,8}.

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

- 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.

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