Infinite-order apeirogonal tiling

In geometry, 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.

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

Symmetry

This tiling represents the fundamental domains of *∞∞ symmetry.

Uniform colorings

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

Domains 0 1 2

symmetry:
[(∞,∞,∞)]  
t0{(∞,∞,∞)}
t1{(∞,∞,∞)}
t2{(∞,∞,∞)}

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

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.
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