Infinite-order apeirogonal tiling

Short description: Pattern on the hyperbolic plane

Infinite-order apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	∞∞
Schläfli symbol	{∞,∞}
Wythoff symbol	∞ | ∞ 2 ∞ ∞ | ∞
Coxeter diagram
Symmetry group	[∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞)
Dual	self-dual
Properties	Vertex-transitive, edge-transitive, face-transitive

The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.

Symmetry

This tiling represents the fundamental domains of *∞^∞ symmetry.

Uniform colorings

This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.

Domains	0	1	2
120px symmetry: [(∞,∞,∞)]	120px t0{(∞,∞,∞)}	120px t1{(∞,∞,∞)}	120px t2{(∞,∞,∞)}

Related polyhedra and tiling

The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain.

240px

a{∞,∞} or = ∪

Paracompact uniform tilings in [∞,∞] family
= =	= =	= =	= =	= =	=	=
{∞,∞}	t{∞,∞}	r{∞,∞}	2t{∞,∞}=t{∞,∞}	2r{∞,∞}={∞,∞}	rr{∞,∞}	tr{∞,∞}
Dual tilings
V∞∞	V∞.∞.∞	V(∞.∞)2	V∞.∞.∞	V∞∞	V4.∞.4.∞	V4.4.∞
Alternations
[1+,∞,∞] (*∞∞2)	[∞+,∞] (∞*∞)	[∞,1+,∞] (*∞∞∞∞)	[∞,∞+] (∞*∞)	[∞,∞,1+] (*∞∞2)	[(∞,∞,2+)] (2*∞∞)	[∞,∞]+ (2∞∞)
h{∞,∞}	s{∞,∞}	hr{∞,∞}	s{∞,∞}	h2{∞,∞}	hrr{∞,∞}	sr{∞,∞}
Alternation duals
V(∞.∞)∞	V(3.∞)3	V(∞.4)4	V(3.∞)3	V∞∞	V(4.∞.4)2	V3.3.∞.3.∞

Paracompact uniform tilings in [(∞,∞,∞)] family
(∞,∞,∞) h{∞,∞}	r(∞,∞,∞) h2{∞,∞}	(∞,∞,∞) h{∞,∞}	r(∞,∞,∞) h2{∞,∞}	(∞,∞,∞) h{∞,∞}	r(∞,∞,∞) r{∞,∞}	t(∞,∞,∞) t{∞,∞}
Dual tilings
V∞∞	V∞.∞.∞.∞	V∞∞	V∞.∞.∞.∞	V∞∞	V∞.∞.∞.∞	V∞.∞.∞
Alternations
[(1+,∞,∞,∞)] (*∞∞∞∞)	[∞+,∞,∞)] (∞*∞)	[∞,1+,∞,∞)] (*∞∞∞∞)	[∞,∞+,∞)] (∞*∞)	[(∞,∞,∞,1+)] (*∞∞∞∞)	[(∞,∞,∞+)] (∞*∞)	[∞,∞,∞)]+ (∞∞∞)
Alternation duals
V(∞.∞)∞	V(∞.4)4	V(∞.∞)∞	V(∞.4)4	V(∞.∞)∞	V(∞.4)4	V3.∞.3.∞.3.∞

See also

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

References

• John Horton 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