Infinite-order apeirogonal tiling

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.

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 vte
= =	= =	= =	= =	= =	=	=

{∞,∞}	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 vte



(∞,∞,∞) 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.∞