Tetraoctagonal tiling

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
Symmetry	[8,4] (*842)	[8,8] = [8,4,1+] (*882) =	[(4,4,4)] = [1+,8,4] (*444) =	[(∞,4,∞,4)] = [1+,8,4,1+] (*4242) = or
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		=	=	= or

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.

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 (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. 

External links

• Weisstein, Eric W.. "Hyperbolic tiling". http://mathworld.wolfram.com/HyperbolicTiling.html. 
• Weisstein, Eric W.. "Poincaré hyperbolic disk". http://mathworld.wolfram.com/PoincareHyperbolicDisk.html. 
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch