Order-4 apeirogonal tiling

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Order-4 apeirogonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex figure∞.∞.∞.∞
Schläfli symbol{∞,4}
r{∞,∞}
t(∞,∞,∞)
t0,1,2,3(∞,∞,∞,∞)
Wythoff symbol4 | ∞ 2
2 | ∞ ∞
∞ ∞ | ∞
Coxeter diagram

Symmetry group[∞,4], (*∞42)
[∞,∞], (*∞∞2)
[(∞,∞,∞)], (*∞∞∞)
(*∞∞∞∞)
DualInfinite-order square tiling
PropertiesVertex-transitive, edge-transitive, face-transitive edge-transitive

In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.

Symmetry

The dual to this tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices.

Uniform colorings

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 bicolor rectified, t1{∞,∞}, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations.

Regular Rectification Fundamental
domains		 Truncations Omnitruncation

[∞,4], (*∞42)
{∞,4}
[∞,∞], (*∞∞2)
t1{∞,∞}
[(∞,4,4)], (*∞44)
[(∞,∞,∞)], (*∞∞∞)
t0,1(∞,∞,∞)
[(∞,∞,∞)], (*∞∞∞)
t1,2(∞,∞,∞)
[(∞,∞,∞)], (*∞∞∞)
t0,2(∞,∞,∞)
(*∞∞∞∞)
t0,1,2,3(∞,∞,∞,∞)

Related polyhedra and tiling

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.

Spherical Euclidean Hyperbolic tilings

{2,4}
{3,4}
{4,4}
{5,4}
{6,4}
{7,4}
{8,4}
 ...
{∞,4}
Noncompact hyperbolic uniform tilings in [∞,4] family
Symmetry: [∞,4], (*∞42)
{∞,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.∞
Noncompact hyperbolic uniform tilings in [∞,∞] family
Symmetry: [∞,∞], (*∞∞2)
{∞,∞} 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∞∞)
h0{∞,∞} h0,1{∞,∞} h1{∞,∞} h1,2{∞,∞} h2{∞,∞} h0,2{∞,∞} s{∞,∞}
Alternation duals
V(∞.∞)∞ V(3.∞)3 V(∞.4)4 V(3.∞)3 V∞∞ V(4.∞.4)2 V3.3.∞.3.∞
Noncompact hyperbolic uniform tilings in [(∞,∞,∞)] family
Symmetry: [(∞,∞,∞)], (*∞∞∞)
(∞,∞,∞) r(∞,∞,∞) (∞,∞,∞) r(∞,∞,∞) (∞,∞,∞) r(∞,∞,∞) t(∞,∞,∞)
Dual tilings
V∞∞ V∞.∞.∞.∞ V∞∞ V∞.∞.∞.∞ V∞∞ V∞.∞.∞.∞ V∞.∞.∞
Alternations
[(1+,∞,∞,∞)]
(*∞∞∞∞)		 [∞+,∞,∞)]
(∞*∞)		 [∞,1+,∞,∞)]
(*∞∞∞∞)		 [∞,∞+,∞)]
(∞*∞)		 [(∞,∞,∞,1+)]
(*∞∞∞∞)		 [(∞,∞,∞+)]
(∞*∞)		 [∞,∞,∞)]+
(∞∞∞)
h(∞,∞,∞) hr(∞,∞,∞) h(∞,∞,∞) hr(∞,∞,∞) h(∞,∞,∞) hr(∞,∞,∞) s(∞,∞,∞)
Alternation duals
V(∞.∞)∞ V(∞.4)4 V(∞.∞)∞ V(∞.4)4 V(∞.∞)∞ V(∞.4)4 V3.∞.3.∞.3.∞

References

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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. 

See also

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