Trioctagonal tiling

Trioctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(3.8)2
Schläfli symbol	r{8,3} or $\left\{\begin{array}{c}8\\ 3\end{array}\right\}$
Wythoff symbol	2 | 8 3| 3 3 | 4
Coxeter diagram	or File:CDel nodes.png File:CDel label4.pngFile:CDel branch 11.pngFile:CDel split2.png
Symmetry group	[8,3], (*832) [(4,3,3)], (*433)
Dual	Order-8-3 rhombille tiling
Properties	Vertex-transitive edge-transitive

In geometry, the trioctagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 octagonal tiling. There are two triangles and two octagons alternating on each vertex. It has Schläfli symbol of r{8,3}.

Symmetry

File:H2 tiling 334-3.png The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles, by Coxeter diagram File:CDel label4.pngFile:CDel branch 11.pngFile:CDel split2.png.

File:Uniform dual tiling 433-t01.png Dual tiling

Related polyhedra and tilings

From a Wythoff construction there are eight hyperbolic 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. Template:Octagonal tiling table

It can also be generated from the (4 3 3) hyperbolic tilings:

Template:Order 4-3-3 tiling table

The trioctagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings: Template:Quasiregular3 table

Template:Quasiregular8 table

See also

• Trihexagonal tiling - 3.6.3.6 tiling
  • Rhombille tiling - dual V3.6.3.6 tiling
• Tilings of regular polygons
• List of uniform tilings

References

• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 Search this book on . (Chapter 19, The Hyperbolic Archimedean Tessellations)
• "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678. Search this book on

External links

• Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
• Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch

Template:Hyperbolic-geometry-stub

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