Truncated order-4 apeirogonal tiling

Poincaré disk model of the hyperbolic plane
Type | Hyperbolic uniform tiling
Vertex configuration | 4.∞.∞
Schläfli symbol | t{∞,4}
tr{∞,∞} or  t { ∞ ∞ } {\displaystyle t{\begin{Bmatrix}\infty \\\\\infty \end{Bmatrix}}}
Wythoff symbol | 2 4 | ∞
2 ∞ ∞ |
Coxeter diagram |

Symmetry group | [∞,4], (*∞42)
[∞,∞], (*∞∞2)
Dual | Infinite-order tetrakis square tiling
Properties | Vertex-transitive

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

## Uniform colorings[edit]

A half symmetry coloring is tr{∞,∞}, has two types of apeirogons, shown red and yellow here. If the apeirogonal curvature is too large, it doesn't converge to a single ideal point, like the right image, red apeirogons below. Coxeter diagram are shown with dotted lines for these divergent, ultraparallel mirrors.

(Vertex centered)  |

(Square centered)

|

## Symmetry[edit]

From [∞,∞] symmetry, there are 15 small index subgroup 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 symmetry can be doubled as ∞42 symmetry by adding a mirror bisecting the fundamental domain. The subgroup index-8 group, [1+,∞,1+,∞,1+] (∞∞∞∞) is the commutator subgroup of [∞,∞].

Small index subgroups of [∞,∞] (*∞∞2)  Index | 1  | 2  | 4

|
|
|

Diagram  |  |  |  |  |  |
Coxeter | [∞,∞]
=  | [1+,∞,∞]
=  | [∞,∞,1+]
=  | [∞,1+,∞]
=  | [1+,∞,∞,1+]
=  | [∞+,∞+]

Orbifold | *∞∞2  | *∞∞∞ | *∞2∞2 | *∞∞∞∞ | ∞∞×
Semidirect subgroups
Diagram  |  |  |  |  |  |
Coxeter  |  | [∞,∞+]
| [∞+,∞]
| [(∞,∞,2+)]
| [∞,1+,∞,1+]
=  =
=  =  | [1+,∞,1+,∞]
=  =
=  =
Orbifold  |  | ∞*∞  | 2*∞∞  | ∞*∞∞
Direct subgroups
Index  | 2  | 4  | 8
Diagram  |  |  |  |  |
Coxeter  | [∞,∞]+
=  | [∞,∞+]+
=  | [∞+,∞]+
=  | [∞,1+,∞]+
=  | [∞+,∞+]+ = [1+,∞,1+,∞,1+]
=  =  =
Orbifold  | ∞∞2  | ∞∞∞  | ∞2∞2  | ∞∞∞∞
Radical subgroups
Index  |  | ∞  | ∞
Diagram  |  |  |  |  |
Coxeter  |  | [∞,∞*]
| [∞*,∞]
| [∞,∞*]+
| [∞*,∞]+

Orbifold  |  | *∞∞ | ∞∞

## Related polyhedra and tiling[edit]

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

* v
* t
* e

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.∞.∞
Paracompact uniform tilings in [∞,4] family

* v
* t
* e

|  |  |  |  |  |
|  |  |  |  |  |
{∞,4} | t{∞,4} | r{∞,4} | 2t{∞,4}=t{4,∞} | 2r{∞,4}={4,∞} | rr{∞,4} | tr{∞,4}
Dual figures
|  |  |  |  |  |
|  |  |  |  |  |
V∞4 | V4.∞.∞ | V(4.∞)2 | V8.8.∞ | V4∞ | V43.∞ | V4.8.∞
Alternations
[1+,∞,4]
(*44∞)  | [∞+,4]
(∞*2)  | [∞,1+,4]
(*2∞2∞)  | [∞,4+]
(4*∞)  | [∞,4,1+]
(*∞∞2)  | [(∞,4,2+)]
(2*2∞)  | [∞,4]+
(∞42)

=  |  |  |  |
=  |  |
h{∞,4}  | s{∞,4} | hr{∞,4}  | s{4,∞}  | h{4,∞}  | hrr{∞,4}  | s{∞,4}
|  |  |  |  |  |
Alternation duals
|  |  |  |  |  |
|  |  |  |  |  |
V(∞.4)4 | V3.(3.∞)2 | V(4.∞.4)2 | V3.∞.(3.4)2 | V∞∞ | V∞.44 | V3.3.4.3.∞
Paracompact uniform tilings in [∞,∞] family

* v
* t
* e

=  |

=  |

=  |

=  |

=  |
=  |

|  |  |  |  |  |
{∞,∞} | t{∞,∞} | r{∞,∞} | 2t{∞,∞}=t{∞,∞} | 2r{∞,∞}={∞,∞} | rr{∞,∞} | tr{∞,∞}
Dual tilings
|  |  |  |  |  |
|  |  |  |  |  |
V∞∞ | V∞.∞.∞ | V(∞.∞)2 | V∞.∞.∞ | V∞∞ | V4.∞.4.∞ | V4.4.∞
Alternations
[1+,∞,∞]
(*∞∞2)  | [∞+,∞]
(∞*∞)  | [∞,1+,∞]
(*∞∞∞∞)  | [∞,∞+]
(∞*∞)  | [∞,∞,1+]
(*∞∞2)  | [(∞,∞,2+)]
(2*∞∞)  | [∞,∞]+
(2∞∞)
|  |  |  |  |  |
|  |  |  |  |  |
h{∞,∞} | s{∞,∞}  | hr{∞,∞}  | s{∞,∞}  | h2{∞,∞} | hrr{∞,∞}  | sr{∞,∞}
Alternation duals
|  |  |  |  |  |
|  |  |  |  |  |
V(∞.∞)∞ | V(3.∞)3 | V(∞.4)4 | V(3.∞)3 | V∞∞ | V(4.∞.4)2 | V3.3.∞.3.∞

## See also[edit]

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

* Uniform tilings in hyperbolic plane
* List of regular polytopes

## References[edit]

* John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.

## External links[edit]

* Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
* Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
* Hyperbolic and Spherical Tiling Gallery

* v
* t
* e

Tessellation

Periodic

* Pythagorean
* Rhombille
* Schwarz triangle
* Rectangle
* Domino
* Uniform tiling and honeycomb
* Coloring
* Convex
* Kisrhombille
* Wallpaper group
* Wythoff

Aperiodic

* Ammann–Beenker
* Aperiodic set of prototiles
* List
* Einstein problem
* Socolar–Taylor
* Gilbert
* Penrose
* Pentagonal
* Pinwheel
* Quaquaversal
* Rep-tile and Self-tiling
* Sphinx
* Socolar
* Truchet

Other

* Anisohedral and Isohedral
* Architectonic and catoptric
* Circle Limit III
* Computer graphics
* Honeycomb
* Isotoxal
* List
* Packing
* Problems
* Domino
* Wang
* Heesch's
* Squaring
* Dividing a square into similar rectangles
* Prototile
* Conway criterion
* Girih
* Regular Division of the Plane
* Regular grid
* Substitution
* Voronoi
* Voderberg

By vertex type

| Spherical|

* 2n
* 33.n
* V33.n
* 42.n
* V42.n

|

Regular|

* 2∞
* 36
* 44
* 63

Semi-
regular|

* 32.4.3.4
* V32.4.3.4
* 33.42
* 33.∞
* 34.6
* V34.6
* 3.4.6.4
* (3.6)2
* 3.122
* 42.∞
* 4.6.12
* 4.82

Hyper-
bolic|

* 32.4.3.5
* 32.4.3.6
* 32.4.3.7
* 32.4.3.8
* 32.4.3.∞
* 32.5.3.5
* 32.5.3.6
* 32.6.3.6
* 32.6.3.8
* 32.7.3.7
* 32.8.3.8
* 33.4.3.4
* 32.∞.3.∞
* 34.7
* 34.8
* 34.∞
* 35.4
* 37
* 38
* 3∞
* (3.4)3
* (3.4)4
* 3.4.62.4
* 3.4.7.4
* 3.4.8.4
* 3.4.∞.4
* 3.6.4.6
* (3.7)2
* (3.8)2
* 3.142
* 3.162
* (3.∞)2
* 3.∞2
* 42.5.4
* 42.6.4
* 42.7.4
* 42.8.4
* 42.∞.4
* 45
* 46
* 47
* 48
* 4∞
* (4.5)2
* (4.6)2
* 4.6.12
* 4.6.14
* V4.6.14
* 4.6.16
* V4.6.16
* 4.6.∞
* (4.7)2
* (4.8)2
* 4.8.10
* V4.8.10
* 4.8.12
* 4.8.14
* 4.8.16
* 4.8.∞
* 4.102
* 4.10.12
* 4.122
* 4.12.16
* 4.142
* 4.162
* 4.∞2
* (4.∞)2
* 54
* 55
* 56
* 5∞
* 5.4.6.4
* (5.6)2
* 5.82
* 5.102
* 5.122
* 64
* 65
* 66
* 68
* 6.4.8.4
* (6.8)2
* 6.82
* 6.102
* 6.122
* 6.162
* 73
* 74
* 77
* 7.62
* 7.82
* 7.142
* 83
* 84
* 86
* 88
* 8.62
* 8.122
* 8.162
* ∞3
* ∞4
* ∞5
* ∞∞
* ∞.62
* ∞.82

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