Truncated infinite-order square tiling

In geometry, the truncated infinite-order square tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{4,∞}.

Uniform color

In (*∞44) symmetry this tiling has 3 colors. Bisecting the isosceles triangle domains can double the symmetry to *∞42 symmetry.

Symmetry

The dual of the tiling represents the fundamental domains of (*∞44) orbifold symmetry. From [(∞,4,4)] (*∞44) symmetry, there are 15 small index subgroup (11 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. The symmetry can be doubled to *∞42 by adding a bisecting mirror across the fundamental domains. The subgroup index-8 group, [(1⁺,∞,1⁺,4,1⁺,4)] (∞22∞22) is the commutator subgroup of [(∞,4,4)].

Small index subgroups of [(∞,4,4)] (*∞44)

Fundamental domains
Subgroup index	1	2			4
Coxeter (orbifold)	[(4,4,∞)] (*∞44)	[(1+,4,4,∞)] (*∞424)	[(4,4,1+,∞)] (*∞424)	[(4,1+,4,∞)] (*∞2∞2)	[(4,1+,4,1+,∞)] 2*∞2∞2	[(1+,4,4,1+,∞)] (∞*2222)
		[(4,4+,∞)] (4*∞2)	[(4+,4,∞)] (4*∞2)	[(4,4,∞+)] (∞*22)	[(1+,4,1+,4,∞)] 2*∞2∞2	[(4+,4+,∞)] (∞22×)
Rotational subgroups
Subgroup index	2	4			8
Coxeter (orbifold)	[(4,4,∞)]+ (∞44)	[(1+,4,4+,∞)] (∞323)	[(4+,4,1+,∞)] (∞424)	[(4,1+,4,∞+)] (∞434)	[(1+,4,1+,4,1+,∞)] = [(4+,4+,∞+)] (∞22∞22)

See also

• Uniform tilings in hyperbolic plane
• 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