View source for Order-6 octagonal tiling

{{Uniform hyperbolic tiles db|Reg hyperbolic tiling stat table|U86_0}}
In [[geometry]], the '''order-6 octagonal tiling''' is a [[List_of_regular_polytopes#Hyperbolic_tilings|regular]] tiling of the [[Hyperbolic geometry|hyperbolic plane]]. It has [[Schläfli symbol]] of {8,6}.
== Symmetry ==
This tiling represents a hyperbolic [[kaleidoscope]] of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by [[orbifold notation]] is called *33333333 with 8 order-3 mirror intersections. In [[Coxeter notation]] can be represented as [8*,6], removing two of three mirrors (passing through the octagon center) in the [[862 symmetry|[8,6] symmetry]].
== Uniform constructions ==
There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] [[kaleidoscope]]. Removing the mirror between the order 2 and 6 points, [8,6,1<sup>+</sup>], gives [(8,8,3)], (*883). Removing two mirrors as [8,6<sup>*</sup>], leaves remaining mirrors (*444444).
{| class=wikitable
|+ Four uniform constructions of 8.8.8.8
|- align=center
!Uniform<BR>Coloring
|[[File:H2 tiling 268-1.png|100px]]
|[[File:H2_tiling_288-2.png|100px]]
|[[File:H2_tiling_688-5.png|100px]]
|
|- align=center valign=top
![[Hyperbolic symmetry|Symmetry]]
|[8,6]<BR>(*862)<BR>{{CDD|node_c1|8|node_c2|6|node_c3}}
|[8,6,1<sup>+</sup>] = [(8,8,3)]<BR>(*883)<BR>{{CDD|node_c1|8|node_c2|6|node_h0}} = {{CDD|node_c1|split1-88|branch_c2}}
|[8,1<sup>+</sup>,6]<BR>(*4232)<BR>{{CDD|node_c1|8|node_h0|6|node_c2}} = {{CDD|label4|branch_c1|2a2b-cross|branch_c2}}
|[8,6<sup>*</sup>]<BR>(*444444)<BR>{{CDD|node_c1|8|node_g|6sg|node_g}}
|- align=center
!Symbol
|{8,6}
|{8,6}{{frac|1|2}}
|r(8,6,8)
|
|- align=center
![[Coxeter diagram|Coxeter<BR>diagram]]
|{{CDD|node_1|8|node|6|node}}
|{{CDD|node_1|8|node|6|node_h0}} = {{CDD|node_1|split1-88|branch}}
|{{CDD|node_1|8|node_h0|6|node}} = {{CDD|label4|branch_11|2a2b-cross|branch}}
|{{CDD|node_1|8|node_g|6sg|node_g}}
|}
== Related polyhedra and tiling ==
This tiling is topologically related as a part of sequence of regular tilings with [[octagon]]al faces, starting with the [[octagonal tiling]], with [[Schläfli symbol]] {8,n}, and [[Coxeter diagram]] {{CDD|node_1|8|node|n|node}}, progressing to infinity.
{{Octagonal tilings}}
{{Order-6 regular tilings}}
{{Order_8-6_tiling_table}}
==See also==
{{Commonscat|Order-6 octagonal 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:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Isohedral tilings]]
[[Category:Order-6 tilings]]
[[Category:Regular tilings]]
[[Category:Octagonal tilings]]