Truncated tetraapeirogonal tiling

In geometry, the truncated tetraapeirogonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one apeirogon on each vertex. It has Schläfli symbol of tr{∞,4}.

Symmetry

The dual of this tiling represents the fundamental domains of [∞,4], (*∞42) symmetry. There are 15 small index subgroups constructed from [∞,4] by mirror removal and alternation. 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 subgroup index-8 group, [1⁺,∞,1⁺,4,1⁺] (∞2∞2) is the commutator subgroup of [∞,4].

A larger subgroup is constructed as [∞,4*], index 8, as [∞,4⁺], (4*∞) with gyration points removed, becomes (*∞∞∞∞) or (*∞⁴), and another [∞*,4], index ∞ as [∞⁺,4], (∞*2) with gyration points removed as (*2^∞). And their direct subgroups [∞,4*]⁺, [∞*,4]⁺, subgroup indices 16 and ∞ respectively, can be given in orbifold notation as (∞∞∞∞) and (2^∞).

Small index subgroups of [∞,4], (*∞42)
Index	1	2			4
Diagram
Coxeter	[∞,4]	[1+,∞,4] =	[∞,4,1+] =	[∞,1+,4] =	[1+,∞,4,1+] =	[∞+,4+]
Orbifold	*∞42	*∞44	*∞∞2	*∞222	*∞2∞2	∞2×
Semidirect subgroups
Diagram
Coxeter		[∞,4+]	[∞+,4]	[(∞,4,2+)]	[1+,∞,1+,4] = = = =	[∞,1+,4,1+] = = = =
Orbifold		4*∞	∞*2	2*∞2	∞*22	2*∞∞
Direct subgroups
Index	2	4			8
Diagram
Coxeter	[∞,4]+ =	[∞,4+]+ =	[∞+,4]+ =	[∞,1+,4]+ =	[∞+,4+]+ = [1+,∞,1+,4,1+] = = =
Orbifold	∞42	∞44	∞∞2	∞222	∞2∞2
Radical subgroups
Index		8	∞	16	∞
Diagram
Coxeter		[∞,4*] =	[∞*,4]	[∞,4*]+ =	[∞*,4]+
Orbifold		*∞∞∞∞	*2∞	∞∞∞∞	2∞

See also

• Tilings of regular polygons
• List of uniform planar tilings

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