Truncated order-4 hexagonal 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.

## Contents

* 1 Constructions
* 2 Dual tiling
* 3 Related polyhedra and tiling
* 3.1 Symmetry
* 4 References
* 5 See also
* 6 External links

## Constructions

There are two uniform constructions of this tiling, first from [6,4] kaleidoscope, and a lower symmetry by removing the last mirror, [6,4,1+], gives [6,6], (*662).

Two uniform constructions of 4.6.4.6  Name  | Tetrahexagonal  | Truncated hexahexagonal

|
|

Image  |  |
Symmetry  | [6,4]
(*642)
| [6,6] = [6,4,1+]
(*662)

Symbol  | t{6,4}  | tr{6,6}
Coxeter diagram  |  |

## Dual tiling

|

The dual tiling, order-6 tetrakis square tiling has face configuration V4.12.12, and represents the fundamental domains of the [6,6] symmetry group.

## Related polyhedra and tiling

### Symmetry

Truncated order-4 hexagonal tiling with *662 mirror lines

The dual of the tiling represents the fundamental domains of (*662) orbifold symmetry. From [6,6] (*662) symmetry, there are 15 small index subgroup (12 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 subgroup index-8 group, [1+,6,1+,6,1+] (3333) is the commutator subgroup of [6,6].

Larger subgroup constructed as [6,6*], removing the gyration points of (6*3), index 12 becomes (*333333).

The symmetry can be doubled to 642 symmetry by adding a mirror to bisect the fundamental domain.

Small index subgroups of [6,6] (*662)

Index  | 1  | 2  | 4
Diagram  |  |  |  |  |  |
Coxeter  | [6,6]
| [1+,6,6]
=  | [6,6,1+]
=  | [6,1+,6]
=  | [1+,6,6,1+]
=  | [6+,6+]

Orbifold  | *662  | *663  | *3232  | *3333  | 33×
Direct subgroups
Diagram  |  |  |  |  |  |
Coxeter  |  | [6,6+]
| [6+,6]
| [(6,6,2+)]
| [6,1+,6,1+]
=  =
=  =  | [1+,6,1+,6]
=  =
=  =
Orbifold  |  | 6*3  | 2*33  | 3*33
Direct subgroups
Index  | 2  | 4  | 8
Diagram  |  |  |  |  |
Coxeter  | [6,6]+
| [6,6+]+
=  | [6+,6]+
=  | [6,1+,6]+
=  | [6+,6+]+ = [1+,6,1+,6]+
=  =  =
Orbifold  | 662  | 663  | 3232  | 3333
Radical subgroups
Index  |  | 12  | 24
Diagram  |  |  |  |  |
Coxeter  |  | [6,6*]
| [6*,6]
| [6,6*]+
| [6*,6]+

Orbifold  |  | *333333  | 333333

## References

* 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.

## See also

* Square tiling
* Tilings of regular polygons
* List of uniform planar tilings
* List of regular polytopes

## External links

* Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html.
* Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html.
* Hyperbolic and Spherical Tiling Gallery
* KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
* Hyperbolic Planar Tessellations, Don Hatch

* v
* t
* e

Tessellation

Periodic

* Pythagorean
* Rhombille
* Schwarz triangle
* Rectangle
* Domino
* Uniform tiling & 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 & Self-tiling
* Sphinx
* Truchet

Other

* Anisohedral & Isohedral
* Architectonic & catoptric
* Circle Limit III
* Computer graphics
* Honeycomb
* Isotoxal
* List
* Packing
* Problems
* Domino
* Wang
* Heesch's
* Squaring
* 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
* (5.∞)2
* 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
* 812
* 8.62
* 8.162
* ∞3
* ∞4
* ∞5
* ∞∞
* ∞.62
* ∞.82

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