Tetraoctagonal tiling

Tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic uniform tiling
Vertex configuration	(4.8)2
Schläfli symbol	r{8,4} or $\left\{\begin{array}{c}8\\ 4\end{array}\right\}$ rr{8,8} rr(4,4,4) t0,1,2,3(∞,4,∞,4)
Wythoff symbol	2 | 8 4
Coxeter diagram	or or
Symmetry group	[8,4], (*842) [8,8], (*882) [(4,4,4)], (*444) [(∞,4,∞,4)], (*4242)
Dual	Order-8-4 quasiregular rhombic tiling
Properties	Vertex-transitive edge-transitive

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions

There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1⁺], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1⁺,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1⁺,8,4,1⁺], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

Four uniform constructions of 4.8.4.8

Name	Tetra-octagonal tiling	Rhombi-octaoctagonal tiling
Image		File:Uniform tiling 88-t02.png	File:Uniform tiling 444-t01.png	File:4242-uniform tiling-verf4848.png
Symmetry	[8,4] (*842) File:CDel node c1.pngError creating thumbnail: File:CDel node c2.pngError creating thumbnail: File:CDel node c3.png	[8,8] = [8,4,1+] (*882) File:CDel node c1.pngError creating thumbnail: File:CDel node c2.pngError creating thumbnail: File:CDel node h0.png = File:CDel node c1.pngError creating thumbnail: File:CDel nodeab c2.png	[(4,4,4)] = [1+,8,4] (*444) File:CDel node h0.pngError creating thumbnail: File:CDel node c2.pngError creating thumbnail: File:CDel node c3.png = Error creating thumbnail: File:CDel branch c2.pngError creating thumbnail: File:CDel node c3.png	[(∞,4,∞,4)] = [1+,8,4,1+] (*4242) File:CDel node h0.pngError creating thumbnail: File:CDel node c2.pngError creating thumbnail: File:CDel node h0.png = Error creating thumbnail: File:CDel branch c2.pngError creating thumbnail: File:CDel branch c2.pngError creating thumbnail: or File:CDel nodeab c2.pngFile:CDel 4a4b-cross.pngFile:CDel nodeab c2.png
Schläfli	r{8,4}	rr{8,8} =r{8,4}1/2	r(4,4,4) =r{4,8}1/2	t0,1,2,3(∞,4,∞,4) =r{8,4}1/4
Coxeter	Error creating thumbnail: Error creating thumbnail:	Error creating thumbnail: Error creating thumbnail: File:CDel node h0.png = Error creating thumbnail: Error creating thumbnail:	File:CDel node h0.pngError creating thumbnail: Error creating thumbnail: = Error creating thumbnail: Error creating thumbnail: Error creating thumbnail:	File:CDel node h0.pngError creating thumbnail: Error creating thumbnail: File:CDel node h0.png = Error creating thumbnail: Error creating thumbnail: Error creating thumbnail: Error creating thumbnail: Error creating thumbnail: or Error creating thumbnail: File:CDel 4a4b-cross.pngError creating thumbnail:

Symmetry

The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.

File:Ord84 qreg rhombic til.png

File:H2chess 248e.png

Related polyhedra and tiling

Template:Quasiregular4 table Template:Quasiregular8 table Template:Order 8-4 tiling table Template:Order 8-8 tiling table Template:Order 4-4-4 tiling table

See also

• Square tiling
• Tilings of regular polygons
• List of uniform planar tilings
• List of regular polytopes

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

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