Infinite-order apeirogonal tiling

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

In geometry, the infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has an infinite number of 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
symmetry: [(∞,∞,∞)]	t0{(∞,∞,∞)}	t1{(∞,∞,∞)}	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.

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
(∞,∞,∞)	r(∞,∞,∞)	(∞,∞,∞)	r(∞,∞,∞)	(∞,∞,∞)	r(∞,∞,∞)	t(∞,∞,∞)
Dual tilings
V∞^∞	V∞.∞.∞.∞	V∞^∞	V∞.∞.∞.∞	V∞^∞	V∞.∞.∞.∞	V∞.∞.∞
Alternations
[(1^+,∞,∞,∞)] (*∞∞∞∞)	[∞^+,∞,∞)] (∞*∞)	[∞,1^+,∞,∞)] (*∞∞∞∞)	[∞,∞^+,∞)] (∞*∞)	[(∞,∞,∞,1^+)] (*∞∞∞∞)	[(∞,∞,∞^+)] (∞*∞)	[∞,∞,∞)]^+ (∞∞∞)
h(∞,∞,∞)	hr(∞,∞,∞)	h(∞,∞,∞)	hr(∞,∞,∞)	h(∞,∞,∞)	hr(∞,∞,∞)	s(∞,∞,∞)
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 H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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. LCCN 99035678.

External links

• Weisstein, Eric W., "Hyperbolic tiling", MathWorld.
• Weisstein, Eric W., "Poincaré hyperbolic disk", MathWorld.
• Hyperbolic and Spherical Tiling Gallery
• KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
• Hyperbolic Planar Tessellations, Don Hatch