Infinite-order square tiling

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

In geometry, the infinite-order square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

There is a half symmetry form, , seen with alternating colors:

This tiling represents the mirror lines of *∞∞∞∞ symmetry. The dual to this tiling defines the fundamental domains of (*2^∞) orbifold symmetry.

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4ⁿ).

*n42 symmetry mutation of regular tilings: {4,n} v t e
Spherical	Euclidean	Compact hyperbolic				Paracompact
{4,3}	{4,4}	{4,5}	{4,6}	{4,7}	{4,8}...	{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.∞

Wikimedia Commons has media related to Infinite-order square tiling.

• Square tiling
• Uniform tilings in hyperbolic plane
• List of regular polytopes

• John H. Conway; Heidi Burgiel; Chaim Goodman-Strauss (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations". The Symmetries of Things. ISBN 978-1-56881-220-5.
• H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 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 Archived 2013-03-24 at the Wayback Machine