Snub tetrahexagonal tiling

Poincaré disk model of the hyperbolic plane
Type | Hyperbolic uniform tiling
Vertex configuration | 3.3.4.3.6
Schläfli symbol | sr{6,4} or  s { 6 4 } {\displaystyle s{\begin{Bmatrix}6\\\4\end{Bmatrix}}}
Wythoff symbol | | 6 4 2
Coxeter diagram |  or
Symmetry group | [6,4]+, (642)
Dual | Order-6-4 floret pentagonal tiling
Properties | Vertex-transitive Chiral

In geometry, the snub tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{6,4}.

## Contents

* 1 Images
* 2 Related polyhedra and tiling
* 3 References
* 4 See also
* 5 External links

## Images[edit]

Drawn in chiral pairs, with edges missing between black triangles:

## Related polyhedra and tiling[edit]

The snub tetrahexagonal tiling  is fifth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n

* v
* t
* e

Symmetry
4n2 | Spherical | Euclidean | Compact hyperbolic  | Paracomp.
242  | 342  | 442  | 542  | 642  | 742  | 842  | ∞42
Snub
figures  |  |  |  |  |  |  |  |
Config. | 3.3.4.3.2 | 3.3.4.3.3 | 3.3.4.3.4 | 3.3.4.3.5 | 3.3.4.3.6 | 3.3.4.3.7 | 3.3.4.3.8 | 3.3.4.3.∞
Gyro
figures  |  |  |  |  |  |  |  |
Config. | V3.3.4.3.2 | V3.3.4.3.3 | V3.3.4.3.4 | V3.3.4.3.5 | V3.3.4.3.6  | V3.3.4.3.7  | V3.3.4.3.8  | V3.3.4.3.∞
Uniform tetrahexagonal tilings

* v
* t
* e

Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)

=  |
=  |

=  |

=  |

=  |

=  |
|  |  |  |  |  |
{6,4} | t{6,4} | r{6,4} | t{4,6} | {4,6} | rr{6,4} | tr{6,4}
Uniform duals
|  |  |  |  |  |
|  |  |  |  |  |
V64 | V4.12.12  | V(4.6)2 | V6.8.8  | V46 | V4.4.4.6  | V4.8.12
Alternations
[1+,6,4]
(*443)  | [6+,4]
(6*2)  | [6,1+,4]
(*3222)  | [6,4+]
(4*3)  | [6,4,1+]
(*662)  | [(6,4,2+)]
(2*32)  | [6,4]+
(642)

=  |
=  |
=  |
=  |
=  |
=  |
|  |  |  |  |  |
h{6,4} | s{6,4} | hr{6,4} | s{4,6} | h{4,6} | hrr{6,4} | sr{6,4}

## References[edit]

* 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.

## See also[edit]

Wikimedia Commons has media related to Uniform tiling 3-3-4-3-6.

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

## External links[edit]

* Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
* Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
* 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 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
* 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
* 8.62
* 8.122
* 8.162
* ∞3
* ∞4
* ∞5
* ∞∞
* ∞.62
* ∞.82

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