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Truncated triapeirogonal tiling | |
---|---|

Poincaré disk model of the hyperbolic plane | |

Type | Hyperbolic uniform tiling |

Vertex configuration | 4.6.∞ |

Schläfli symbol | tr{∞,3} or |

Wythoff symbol | 2 ∞ 3 | |

Coxeter diagram | |

Symmetry group | [∞,3], (*∞32) |

Dual | Order 3-infinite kisrhombille |

Properties | Vertex-transitive |

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

**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 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.

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

The dual of this tiling represents the fundamental domains of [∞,3], *∞32 symmetry. There are 3 small index subgroup constructed from [∞,3] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.

A special index 4 reflective subgroup, is [(∞,∞,3)], (*∞∞3), and its direct subgroup [(∞,∞,3)]^{+}, (∞∞3), and semidirect subgroup [(∞,∞,3^{+})], (3*∞).^{ [1] } Given [∞,3] with generating mirrors {0,1,2}, then its index 4 subgroup has generators {0,121,212}.

An index 6 subgroup constructed as [∞,3*], becomes [(∞,∞,∞)], (*∞∞∞).

Index | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 24 | ||
---|---|---|---|---|---|---|---|---|---|---|

Diagrams | ||||||||||

Coxeter (orbifold) | [∞,3] (*∞32) | [1^{+},∞,3](*∞33) | [∞,3^{+}](3*∞) | [∞,∞] (*∞∞2) | [(∞,∞,3)] (*∞∞3) | [∞,3*] (*∞ ^{3}) | [∞,1^{+},∞](*(∞2) ^{2}) | [(∞,1^{+},∞,3)](*(∞3) ^{2}) | [1^{+},∞,∞,1^{+}](*∞ ^{4}) | [(∞,∞,3*)] (*∞ ^{6}) |

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

Index | 2 | 4 | 6 | 8 | 12 | 16 | 24 | 48 | ||

Diagrams | ||||||||||

Coxeter (orbifold) | [∞,3]^{+}(∞32) | [∞,3^{+}]^{+}(∞33) | [∞,∞]^{+}(∞∞2) | [(∞,∞,3)]^{+}(∞∞3) | [∞,3*]^{+}(∞ ^{3}) | [∞,1^{+},∞]^{+}(∞2) ^{2} | [(∞,1^{+},∞,3)]^{+}(∞3) ^{2} | [1^{+},∞,∞,1^{+}]^{+}(∞ ^{4}) | [(∞,∞,3*)]^{+}(∞ ^{6}) |

Paracompact uniform tilings in [∞,3] family | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Symmetry: [∞,3], (*∞32) | [∞,3]^{+}(∞32) | [1^{+},∞,3](*∞33) | [∞,3^{+}](3*∞) | |||||||

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

{∞,3} | t{∞,3} | r{∞,3} | t{3,∞} | {3,∞} | rr{∞,3} | tr{∞,3} | sr{∞,3} | h{∞,3} | h_{2}{∞,3} | s{3,∞} |

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

V∞^{3} | V3.∞.∞ | V(3.∞)^{2} | V6.6.∞ | V3^{∞} | V4.3.4.∞ | V4.6.∞ | V3.3.3.3.∞ | V(3.∞)^{3} | V3.3.3.3.3.∞ |

This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram *p*< 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For *p*> 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.

A **zonohedron** is a convex polyhedron with point symmetry, every face of which is a polygon with point symmetry. Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in three-dimensional space, or as the three-dimensional projection of a hypercube. Zonohedra were originally defined and studied by E. S. Fedorov, a Russian crystallographer. More generally, in any dimension, the Minkowski sum of line segments forms a polytope known as a **zonotope**.

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}.

*n32 symmetry mutations of omnitruncated tilings: 4.6.2n | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Sym. * n32 [ n,3] | Spherical | Euclid. | Compact hyperb. | Paraco. | 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] | [3i,3] | |

Figures | ||||||||||||

Config. | 4.6.4 | 4.6.6 | 4.6.8 | 4.6.10 | 4.6.12 | 4.6.14 | 4.6.16 | 4.6.∞ | 4.6.24i | 4.6.18i | 4.6.12i | 4.6.6i |

Duals | ||||||||||||

Config. | V4.6.4 | V4.6.6 | V4.6.8 | V4.6.10 | V4.6.12 | V4.6.14 | V4.6.16 | V4.6.∞ | V4.6.24i | V4.6.18i | V4.6.12i | V4.6.6i |

Wikimedia Commons has media related to . Uniform tiling 4-6-i |

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 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-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 infinite-order triangular tiling** is a uniform tiling of the hyperbolic plane with a Schläfli symbol of t{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 **order-6 pentagonal tiling** is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.

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}.

- ↑ Norman W. Johnson and Asia Ivic Weiss,
*Quadratic Integers and Coxeter Groups*, Can. J. Math. Vol. 51 (6), 1999 pp. 1307–1336

- 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.

- Weisstein, Eric W. "Hyperbolic tiling".
*MathWorld*. - Weisstein, Eric W. "Poincaré hyperbolic disk".
*MathWorld*.

**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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