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.

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(∞.∞)	V∞.∞.∞	V∞	V4.∞.4.∞	V4.4.∞
Alternations
(*∞∞2)	(∞*∞)	(*∞∞∞∞)	(∞*∞)	(*∞∞2)	(2*∞∞)	(2∞∞)


h{∞,∞}	s{∞,∞}	hr{∞,∞}	s{∞,∞}	h₂{∞,∞}	hrr{∞,∞}	sr{∞,∞}
Alternation duals


V(∞.∞)	V(3.∞)	V(∞.4)	V(3.∞)	V∞	V(4.∞.4)	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
(*∞∞∞∞)	(∞*∞)	(*∞∞∞∞)	(∞*∞)	(*∞∞∞∞)	(∞*∞)	(∞∞∞)


Alternation duals

V(∞.∞)	V(∞.4)	V(∞.∞)	V(∞.4)	V(∞.∞)	V(∞.4)	V3.∞.3.∞.3.∞

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. LCCN 99035678.

Periodic
Pythagorean Rhombille Schwarz triangle Rectangle Domino Uniform tiling and honeycomb Coloring Convex Kisrhombille Wallpaper group Wythoff

Aperiodic
Ammann–Beenker Aperiodic set of prototiles List Einstein problem Socolar–Taylor Gilbert Penrose Pinwheel Quaquaversal Rep-tile and Self-tiling Sphinx Socolar Truchet

Other
Anisohedral and Isohedral Architectonic and catoptric Circle Limit III Computer graphics Honeycomb Isotoxal List Packing Pentagonal Problems Domino Wang Heesch's Squaring Dividing a square into similar rectangles Prototile Conway criterion Girih Regular Division of the Plane Regular grid Substitution Voronoi Voderberg