Infinite-order square tiling
Infinite-order square tiling
Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration4
Schläfli symbol{4,}
Wythoff symbol | 4 2
Coxeter diagram
Symmetry group[,4], (*42)
DualOrder-4 apeirogonal tiling
PropertiesVertex-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, , seen with alternating colors:

Symmetry

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}
Spherical Euclidean Compact hyperbolic Paracompact

{4,3}

{4,4}

{4,5}

{4,6}

{4,7}

{4,8}...

{4,}
Paracompact uniform tilings in [,4] family
{,4} t{,4} r{,4} 2t{,4}=t{4,} 2r{,4}={4,} rr{,4} tr{,4}
Dual figures
V4 V4.. V(4.)2 V8.8. V4 V43. V4.8.
Alternations
[1+,,4]
(*44)
[+,4]
(*2)
[,1+,4]
(*22)
[,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.

See also

References

    • 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.
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