Hexaoctagonal tiling

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

Constructions

There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,6] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,6,1⁺], gives [(8,8,3)], (*883). Removing the mirror between the order 2 and 8 points, [1⁺,8,6], gives [(4,6,6)], (*664). Removing two mirrors as [8,1⁺,6,1⁺], leaves remaining mirrors (*4343).

Four uniform constructions of 6.8.6.8

Uniform Coloring
Symmetry	[8,6] (*862)	[(8,3,8)] = [8,6,1+] (*883)	[(6,4,6)] = [1+,8,6] (*664)	[1+,8,6,1+] (*4343)
Symbol	r{8,6}	r{(8,3,8)}	r{(6,4,6)}
Coxeter diagram		=	=	=

Symmetry

The dual tiling has face configuration V6.8.6.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4343), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*43) orbifold. These are subsymmetries of [8,6].

[1+,8,4,1+], (*4343)	[(8,4,2+)], (2*43)

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