Infinite-order apeirogonal tiling

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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.

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Symmetry

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

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

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

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.

a{∞,∞} or = ∪

Paracompact uniform tilings in [∞,∞] family v t e
= =	= =	= =	= =	= =	=	=

{∞,∞}	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 v t e



(∞,∞,∞)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.∞