View source for Order-4 hexagonal tiling

{{short description|Regular tiling of the hyperbolic plane}}
{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U64_0}}
In [[geometry]], the '''order-4 hexagonal tiling''' is a [[List of regular polytopes#Hyperbolic tilings|regular]] tiling of the [[Hyperbolic geometry|hyperbolic plane]]. It has [[Schläfli symbol]] of {6,4}.

== Symmetry ==
This tiling represents a hyperbolic [[kaleidoscope]] of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by [[orbifold notation]] is called *222222 with 6 order-2 mirror intersections. In [[Coxeter notation]] can be represented as [6<sup>*</sup>,4], removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral [[4422 symmetry|*4422 symmetry]]. Adding 3 bisecting mirrors through the vertices defines [[443 symmetry|*443 symmetry]]. Adding 3 bisecting mirrors through the edge defines [[3222 symmetry|*3222 symmetry]]. Adding all 6 bisectors leads to full [[642 symmetry|*642 symmetry]].
{| class=wikitable
|- align=center 
|[[File:642 symmetry zz0.png|80px]]<br>*222222
|[[File:642 symmetry a00.png|80px]]<br>*443
|[[File:642 symmetry 0a0.png|80px]]<br>*3222
|[[File:642 symmetry 000.png|80px]]<br>*642
|}

== Uniform colorings ==
There are 7 distinct [[uniform coloring]]s for the order-4 hexagonal tiling. They are similar to 7 of the [[Square tiling#Uniform colorings|uniform colorings of the square tiling]], but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and [[Coxeter diagram]]s while three of them are undercolorings.

{| class=wikitable
|+ Uniform constructions of 6.6.6.6
|- align=center
!
!1 color
!2 colors
!colspan=2|3 and 2 colors
!colspan=3|4, 3 and 2 colors
|- align=center
!Uniform<br>Coloring
|[[File:H2 tiling 246-1.png|80px]]<br>(1111)
|[[File:H2_tiling_266-2.png|80px]]<br>(1212)
|[[File:H2_tiling_366-5.png|80px]]<br>(1213)
|[[File:H2_tiling_366-5_undercolor.png|80px]]<br>(1113)
|[[File:Order-4_hexagonal_tiling_nonsimplex_domain.png|80px]]<br>(1234)
|[[File:Order-4_hexagonal_tiling_nonsimplex_domain_undercolor.png|80px]]<br>(1123)
|[[File:Order-4_hexagonal_tiling_row_coloring.png|80px]]<br>(1122)
|- align=center valign=top
![[Hyperbolic symmetry|Symmetry]]
|[6,4]<br>([[642 symmetry|*642]])<br>{{CDD|node_c1|6|node_c2|4|node_c3}}
|[6,6]<br>([[662 symmetry|*662]])<br> {{CDD|node_c1|split1-66|nodeab_c2}} = {{CDD|node_c1|6|node_c2|4|node_h0}}
|colspan=2|[(6,6,3)] = [6,6,1<sup>+</sup>]<br>([[663 symmetry|*663]])<br> {{CDD|node_c2|split1-66|branch_c1}} = {{CDD|node_c2|6|node_c1|6|node_h0}}
|colspan=3|[1<sup>+</sup>,6,6,1<sup>+</sup>]<br>([[3333 symmetry|*3333]])<br> {{CDD|branch_c1|3a3b-cross|branch_c1}} = {{CDD|node_h0|6|node_c1|6|node_h0}} = {{CDD|node_c1|6|node_g|4sg|node_g}}
|- align=center
!Symbol
|{6,4}
|r{6,6} = {6,4}<sup>1</sup>/<sub>2</sub>
|colspan=2|r(6,3,6) = r{6,6}<sup>1</sup>/<sub>2</sub>
|colspan=3|r{6,6}<sup>1</sup>/<sub>4</sub>
|- align=center
![[Coxeter diagram|Coxeter<br>diagram]]
|{{CDD|node_1|6|node|4|node}}
|{{CDD|node_1|split1-66|nodes}} = {{CDD|node_1|6|node|4|node_h0}}
|colspan=2|{{CDD|node|split1-66|branch_11}} = {{CDD|node|6|node_1|6|node_h0}} 
|colspan=3|{{CDD|branch_11|3a3b-cross|branch_11}} = {{CDD|node_h0|6|node_1|6|node_h0}} = {{CDD|node_1|6|node_g|4sg|node_g}}
|}

== Regular maps==
The [[Regular map (graph theory)|regular map]] {6,4}<sub>3</sub> or {6,4}<sub>(4,0)</sub> can be seen as a 4-coloring on the {6,4} tiling. It also has a representation as a [[petrial octahedron]], {3,4}<sup>{{pi}}</sup>, an abstract polyhedron with vertices and edges of an [[octahedron]], but instead connected by 4 [[Petrie polygon]] faces.
:[[File:Regular map 6 4-3 pattern.png|320px]][[File:Octahedron_4_petrie_polygons.png|320px]]
== Related polyhedra and tiling ==

This tiling is topologically related as a part of sequence of regular tilings with [[hexagon]]al faces, starting with the [[hexagonal tiling]], with [[Schläfli symbol]] {6,n}, and [[Coxeter diagram]] {{CDD|node_1|6|node|n|node}}, progressing to infinity.
{{Hexagonal_regular_tilings}}

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the [[octahedron]], with [[Schläfli symbol]] {n,4}, and Coxeter diagram {{CDD|node_1|n|node|4|node}}, with n progressing to infinity.
{{Order-4_regular_tilings}}
{{Quasiregular6 table}}

{{Order 6-4 tiling table}}
{{Order 6-6 tiling table}}
{{Order 3-2-3-2 tiling table}}
{{Order_3-2-2-2_tiling_table}}

==See also==
{{Commons category|Order-4 hexagonal tiling}}
*[[Square tiling]]
*[[Tilings of regular polygons]]
*[[List of uniform planar tilings]]
*[[List of regular polytopes]]

==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}}

== 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:Isohedral tilings]]
[[Category:Order-4 tilings]]
[[Category:Regular tilings]]