Infinite-order square tiling


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
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, , 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
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.∞

See also

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

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.