Infinite-order square tiling

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Infinite-order square tiling
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 CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png
CDel node 1.pngCDel split1-44.pngCDel branch.pngCDel labelinfin.png
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.

Uniform colorings[]

There is a half symmetry form, CDel node 1.pngCDel split1-44.pngCDel branch.pngCDel labelinfin.png, seen with alternating colors:

H2 tiling 44i-4.png

Symmetry[]

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

H2chess 24ic.png

Related polyhedra and tiling[]

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
Spherical Euclidean Compact hyperbolic Paracompact
Uniform tiling 432-t0.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png 		Uniform tiling 44-t0.svg
{4,4}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 4.pngCDel node.png 		H2-5-4-primal.svg
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 246-4.png
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png 		H2 tiling 247-4.png
{4,7}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 7.pngCDel node.png 		H2 tiling 248-4.png
{4,8}...
CDel node 1.pngCDel 4.pngCDel node.pngCDel 8.pngCDel node.png 		H2 tiling 24i-4.png
{4,∞}
CDel node 1.pngCDel 4.pngCDel node.pngCDel infin.pngCDel node.png
Paracompact uniform tilings in [∞,4] family
  • v
CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 4.pngCDel node 1.png
H2 tiling 24i-1.png H2 tiling 24i-3.png H2 tiling 24i-2.png H2 tiling 24i-6.png H2 tiling 24i-4.png H2 tiling 24i-5.png H2 tiling 24i-7.png
{∞,4} t{∞,4} r{∞,4} 2t{∞,4}=t{4,∞} 2r{∞,4}={4,∞} rr{∞,4} tr{∞,4}
Dual figures
CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel 4.pngCDel node f1.png
H2chess 24ib.png H2chess 24if.png H2chess 24ia.png H2chess 24ie.png H2chess 24ic.png H2chess 24id.png H2checkers 24i.png
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)
CDel node h1.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png
= CDel branch 10ru.pngCDel split2-44.pngCDel node.png 		CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node h1.png
= CDel node.pngCDel split1-ii.pngCDel nodes 10lu.png 		CDel node h.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node h.png CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png
h{∞,4} s{∞,4} hr{∞,4} s{4,∞} h{4,∞} hrr{∞,4} s{∞,4}
H2 tiling 44i-1.png Uniform tiling i42-h01.png H2 tiling 2ii-1.png Uniform tiling i42-snub.png
Alternation duals
CDel node fh.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node fh.png
H2chess 44ib.png H2 tiling 2ii-4.png
V(∞.4)4 V3.(3.∞)2 V(4.∞.4)2 V3.∞.(3.4)2 V∞∞ V∞.44 V3.3.4.3.∞

See also[]

References[]

  • John H. Conway; Heidi Burgiel; Chaim Goodman-Strass (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.

External links[]

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