View source for Rhombitetraoctagonal tiling

{{Short description|Regular tiling of the hyperbolic plane}}
In [[Geometry|geometry]], the '''rhombitetraoctagonal tiling''' is a uniform tiling of the [[Hyperbolic geometry|hyperbolic plane]]. It has [[Schläfli symbol]] of rr{8,4}. It can be seen as constructed as a [[Rectification (geometry)|rectified]] [[Tetraoctagonal tiling|tetraoctagonal tiling]], r{8,4}, as well as an [[Expansion (geometry)|expanded]] [[Order-4 octagonal tiling|order-4 octagonal tiling]] or expanded [[Order-8 square tiling|order-8 square tiling]].

== Constructions ==
There are two uniform constructions of this tiling, one from [8,4] or (*842) symmetry, and secondly removing the mirror middle, [8,1<sup>+</sup>,4], gives a rectangular fundamental domain [∞,4,∞], (*4222).
{| class=wikitable width=360
|+ Two uniform constructions of 4.4.4.8
|- align=center
!Name
|Rhombitetraoctagonal tiling
|
|- align=center
!Image
|[[File:Uniform tiling 84-t02.png|180px]]
|[[File:Uniform tiling 4.4.4.8.png|180px]]
|- align=center
!Symmetry
|[8,4]<BR>(*842)<BR>{{CDD|node_c1|8|node_c3|4|node_c2}}
|[8,1<sup>+</sup>,4] = [∞,4,∞]<BR>(*4222)<BR>{{CDD|node_c1|8|node_h0|4|node_c2}} = {{CDD|label4|branch_c1|2a2b-cross|nodeab_c2}}
|- align=center
![[Schläfli symbol]]
|rr{8,4}
|t<sub>0,1,2,3</sub>{∞,4,∞}
|- align=center
!Coxeter diagram
|{{CDD|node_1|8|node|4|node_1}}
|{{CDD|node_1|8|node_h0|4|node_1}} = {{CDD|label4|branch_11|2a2b-cross|nodes_11}}
|}

== Symmetry ==
A lower symmetry construction exists, with (*4222) [[Orbifold notation|orbifold]] symmetry. This symmetry can be seen in the dual tiling, called a ''deltoidal tetraoctagonal tiling'', alternately colored here. Its fundamental domain is a [[Lambert quadrilateral]], with 3 right angles. 

{| class=wikitable width=320
|[[File:Deltoidal_tetraoctagonal_til.png|160px]]
|[[File:H2chess_248d.png|160px]]
|-
|colspan=2|The dual tiling, called a ''deltoidal tetraoctagonal tiling'', represents the fundamental domains of the *4222 orbifold.
|}

With edge-colorings there is a half symmetry form (4*4) [[Orbifold notation|orbifold notation]]. The octagons can be considered as truncated squares, t{4} with two types of edges. It has Coxeter diagram {{CDD|node_h|4|node_h|8|node_1}}, [[Schläfli symbol]] s<sub>2</sub>{4,8}. The squares can be distorted into [[Isosceles trapezoid|isosceles trapezoid]]s. In the limit, where the rectangles degenerate into edges, an [[Order-8 square tiling|order-8 square tiling]] results, constructed as a [[Snub tetraoctagonal tiling|snub tetraoctagonal tiling]], {{CDD|node_h|4|node_h|8|node}}.

== Related polyhedra and tiling ==



==References==
* [[Biography: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:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Uniform tilings]]

{{Sourceattribution|Rhombitetraoctagonal tiling}}