View source for Rhombitrioctagonal tiling

{{Short description|Semiregular tiling of the hyperbolic plane}}
{{Uniform hyperbolic tiles db|Uniform hyperbolic tiling stat table|U83_02}}
In [[geometry]], the '''rhombitrioctagonal tiling''' is a semiregular tiling of the
[[Hyperbolic space|hyperbolic plane]]. At each [[vertex (geometry)|vertex]] of the tiling there is one [[triangle]] and one [[octagon]], alternating between two [[square (geometry)|square]]s. The tiling has [[Schläfli symbol]] rr{8,3}. It can be seen as constructed as a [[Rectification (geometry)|rectified]] [[trioctagonal tiling]], r{8,3}, as well as an [[Expansion (geometry)|expanded]] [[octagonal tiling]] or expanded [[order-8 triangular tiling]].
== Symmetry ==
This tiling has [8,3], (*832) symmetry. There is only one uniform coloring.
Similar to the Euclidean [[rhombitrihexagonal tiling]], by edge-coloring there is a half symmetry form (3*4) [[orbifold notation]]. The octagons can be considered as truncated squares, t{4} with two types of edges. It has [[Coxeter diagram]] {{CDD|node_h|3|node_h|8|node_1}}, [[Schläfli symbol]] s<sub>2</sub>{3,8}. The squares can be distorted into [[isosceles trapezoid]]s. In the limit, where the rectangles degenerate into edges, an [[order-8 triangular tiling]] results, constructed as a [[snub tritetratrigonal tiling]], {{CDD|node_h|3|node_h|8|node}}.
== Related polyhedra and tilings ==
From a [[Wythoff construction]] there are ten hyperbolic [[Uniform tilings in hyperbolic plane|uniform tilings]] that can be based from the regular octagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
{{Octagonal tiling table}}
=== Symmetry mutations===
This tiling is topologically related as a part of sequence of [[Cantellation (geometry)|cantellated]] polyhedra with vertex figure (3.4.n.4), and continues as tilings of the [[Hyperbolic space|hyperbolic plane]]. These [[vertex-transitive]] figures have (*n32) reflectional [[Orbifold notation|symmetry]].
{{Expanded table}}
== See also ==
{{Commons category|Uniform tiling 3-4-8-4}}
* [[Rhombitrihexagonal tiling]]
* [[Order-3 octagonal tiling]]
* [[Tilings of regular polygons]]
* [[List of uniform tilings]]
* [[Kagome lattice]]
==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.hadron.org/~hatch/HyperbolicTesselations Hyperbolic Planar Tessellations, Don Hatch]
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[[Category:Hyperbolic tilings]]
[[Category:Isogonal tilings]]
[[Category:Semiregular tilings]]
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