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Infinite-order apeirogonal tiling

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

The infinite-order apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,∞}, which means it has countably infinitely many apeirogons around all its ideal vertices.

Symmetry

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This tiling represents the fundamental domains of *∞ symmetry.

Uniform colorings

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This tiling can also be alternately colored in the [(∞,∞,∞)] symmetry from 3 generator positions.

Domains 0 1 2

symmetry:
[(∞,∞,∞)]  

t0{(∞,∞,∞)}

t1{(∞,∞,∞)}

t2{(∞,∞,∞)}
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The union of this tiling and its dual can be seen as orthogonal red and blue lines here, and combined define the lines of a *2∞2∞ fundamental domain.

a{∞,∞} or =
Paracompact uniform tilings in [∞,∞] family

=
=

=
=

=
=

=
=

=
=

=

=
{∞,∞} 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∞∞)
h{∞,∞} s{∞,∞} hr{∞,∞} s{∞,∞} h2{∞,∞} hrr{∞,∞} sr{∞,∞}
Alternation duals
V(∞.∞) V(3.∞)3 V(∞.4)4 V(3.∞)3 V∞ V(4.∞.4)2 V3.3.∞.3.∞
Paracompact uniform tilings in [(∞,∞,∞)] family
(∞,∞,∞)
h{∞,∞}
r(∞,∞,∞)
h2{∞,∞}
(∞,∞,∞)
h{∞,∞}
r(∞,∞,∞)
h2{∞,∞}
(∞,∞,∞)
h{∞,∞}
r(∞,∞,∞)
r{∞,∞}
t(∞,∞,∞)
t{∞,∞}
Dual tilings
V∞ V∞.∞.∞.∞ V∞ V∞.∞.∞.∞ V∞ V∞.∞.∞.∞ V∞.∞.∞
Alternations
[(1+,∞,∞,∞)]
(*∞∞∞∞)
[∞+,∞,∞)]
(∞*∞)
[∞,1+,∞,∞)]
(*∞∞∞∞)
[∞,∞+,∞)]
(∞*∞)
[(∞,∞,∞,1+)]
(*∞∞∞∞)
[(∞,∞,∞+)]
(∞*∞)
[∞,∞,∞)]+
(∞∞∞)
Alternation duals
V(∞.∞) V(∞.4)4 V(∞.∞) V(∞.4)4 V(∞.∞) V(∞.4)4 V3.∞.3.∞.3.∞

See also

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References

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  • John Horton Conway, Heidi Burgiel, Chaim Goodman-Strauss, 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.
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