View source for Truncated order-6 hexagonal tiling

{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U66_01}}
In [[geometry]], the '''truncated order-6 hexagonal tiling''' is a uniform tiling of the [[Hyperbolic geometry|hyperbolic plane]]. It has [[Schläfli symbol]] of t{6,6}. It can also be identically constructed as a '''cantic order-6 square tiling''', h<sub>2</sub>{4,6}
== Uniform colorings ==
By *663 symmetry, this tiling can be constructed as an [[omnitruncation]], t{(6,6,3)}:
:[[File:H2_tiling_366-7.png|240px]]
== Symmetry ==
[[File:Truncated_order-6_hexagonal_tiling_with_mirrors.png|thumb|left|Truncated order-6 hexagonal tiling  with *663 mirror lines]]
The dual to this tiling represent the fundamental domains of [(6,6,3)] (*663) symmetry. There are 3 small index subgroup symmetries constructed from [(6,6,3)] by mirror removal and alternation. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors.
The symmetry can be doubled as [[662 symmetry]] by adding a mirror bisecting the fundamental domain.
{{-}}
{| class=wikitable
|+ Small index subgroups of [(6,6,3)] (*663)
|- align=center
![[Subgroup index|Index]]
!1
!colspan=2|2
!6
|- align=center
!Diagram
|[[File:663_symmetry_000.png|150px]]
|[[File:663_symmetry_0a0.png|150px]]
|[[File:663_symmetry_a0a.png|150px]]
|[[File:663_symmetry_z0z.png|150px]]
|- align=center
![[Coxeter notation|Coxeter]]<BR>([[Orbifold notation|orbifold]])
|[(6,6,3)] = {{CDD|node_c1|split1-66|branch_c2}}<BR>(*663)
|[(6,1<sup>+</sup>,6,3)] = {{CDD|labelh|node|split1-66|branch_c2}} = {{CDD|branch_c2|3a3b-cross|branch_c2}}<BR>([[3232 symmetry|*3333]])
|[(6,6,3<sup>+</sup>)] = {{CDD|node_c1|split1-66|branch_h2h2}}<BR>(3*33)
|[(6,6,3*)] = {{CDD|node_c1|split1-66|branch|labels}}<BR>([[333333 symmetry|*333333]])
|- align=center
!colspan=5|Direct subgroups
|- align=center
!Index
!2
!colspan=2|4
!12
|- align=center
!Diagram
|[[File:663_symmetry_aaa.png|150px]]
|colspan=2|[[File:663_symmetry_abc.png|150px]]
|[[File:663_symmetry_zaz.png|150px]]
|- align=center
!Coxeter<BR>(orbifold)
|[(6,6,3)]<sup>+</sup> = {{CDD|node_h2|split1-66|branch_h2h2}}<BR>(663)
|colspan=2|[(6,6,3<sup>+</sup>)]<sup>+</sup> = {{CDD|labelh|node|split1-66|branch_h2h2}} = {{CDD|branch_h2h2|3a3b-cross|branch_h2h2}}<BR>(3333)
|[(6,6,3*)]<sup>+</sup> = {{CDD|node_h2|split1-66|branch|labels}}<BR>(333333)
|}
== Related polyhedra and tiling ==
{{Order 6-6 tiling table}}
==References==
* [[John Horton Conway|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)
* {{Cite book|title=The Beauty of Geometry: Twelve Essays|year=1999|publisher=Dover Publications|lccn=99035678|isbn=0-486-40919-8|chapter=Chapter 10: Regular honeycombs in hyperbolic space}}
==See also==
*[[Square tiling]]
*[[Tilings of regular polygons]]
*[[List of uniform planar tilings]]
*[[List of regular polytopes]]
== External links ==
*{{MathWorld | urlname= HyperbolicTiling | title = Hyperbolic tiling}}
*{{MathWorld | urlname=PoincareHyperbolicDisk | title = Poincaré hyperbolic disk }}
* [http://bork.hampshire.edu/~bernie/hyper/ Hyperbolic and Spherical Tiling Gallery]
* [http://geometrygames.org/KaleidoTile/index.html KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings]
* [http://www.plunk.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]
{{Tessellation}}
[[Category:Hexagonal tilings]]
[[Category:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Order-6 tilings]]
[[Category:Truncated tilings]]
[[Category:Uniform tilings]]