Order-4 apeirogonal tiling

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Order-4 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type	Hyperbolic regular tiling
Vertex configuration	∞4
Schläfli symbol	{∞,4} r{∞,∞} t(∞,∞,∞) t0,1,2,3(∞,∞,∞,∞)
Wythoff symbol	4 | ∞ 2 2 | ∞ ∞ ∞ ∞ | ∞
Coxeter diagram
Symmetry group	[∞,4], (*∞42) [∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞) (*∞∞∞∞)
Dual	Infinite-order square tiling
Properties	Vertex-transitive, edge-transitive, face-transitive edge-transitive

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It is a way of covering the hyperbolic plane—a non-Euclidean surface with constant negative curvature—with a repeating pattern of identical shapes without gaps or overlaps.

This tiling is constructed from apeirogons, which are polygons with an infinite number of sides. In this specific pattern, four of these apeirogons meet at each vertex. It can be seen as the hyperbolic equivalent of the familiar square tiling of the Euclidean plane, where four squares meet at each vertex. Its Schläfli symbol is {∞,4}, with the "4" indicating that four polygons meet at a vertex and the "∞" indicating that the polygon used is an apeirogon.

This tiling represents the mirror lines of *2^∞ symmetry. Its dual tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices.

Like the Euclidean square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r{∞,∞}, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations.

1 color	2 color	3 and 2 colors		4, 3 and 2 colors
[∞,4], (*∞42)	[∞,∞], (*∞∞2)	[(∞,∞,∞)], (*∞∞∞)		(*∞∞∞∞)
{∞,4}	r{∞,∞} = {∞,4}1⁄2	t0,2(∞,∞,∞) = r{∞,∞}1⁄2		t0,1,2,3(∞,∞,∞,∞) = r{∞,∞}1⁄4 = {∞,4}1⁄8
(1111)	(1212)	(1213)	(1112)	(1234)	(1123)	(1122)
	=	= =		= =

This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.

*n42 symmetry mutation of regular tilings: {n,4} v t e
Spherical		Euclidean	Hyperbolic tilings
24	34	44	54	64	74	84	...∞4

Paracompact uniform tilings in [∞,4] family v t e
{∞,4}	t{∞,4}	r{∞,4}	2t{∞,4}=t{4,∞}	2r{∞,4}={4,∞}	rr{∞,4}	tr{∞,4}
Dual figures
V∞4	V4.∞.∞	V(4.∞)2	V8.8.∞	V4∞	V43.∞	V4.8.∞
Alternations
[1+,∞,4] (*44∞)	[∞+,4] (∞*2)	[∞,1+,4] (*2∞2∞)	[∞,4+] (4*∞)	[∞,4,1+] (*∞∞2)	[(∞,4,2+)] (2*2∞)	[∞,4]+ (∞42)
=				=
h{∞,4}	s{∞,4}	hr{∞,4}	s{4,∞}	h{4,∞}	hrr{∞,4}	s{∞,4}
Alternation duals
V(∞.4)4	V3.(3.∞)2	V(4.∞.4)2	V3.∞.(3.4)2	V∞∞	V∞.44	V3.3.4.3.∞

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.∞

Wikimedia Commons has media related to Order-4 apeirogonal tiling.

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

• John H. 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.

• 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