Infinite-order apeirogonal tiling
Infinite-order apeirogonal tiling | |
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![]() 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 |
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.
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 |
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![]() symmetry: [(∞,∞,∞)] ![]() ![]() ![]() ![]() | ![]() t0{(∞,∞,∞)} ![]() ![]() ![]() ![]() | ![]() t1{(∞,∞,∞)} ![]() ![]() ![]() ![]() | ![]() t2{(∞,∞,∞)} ![]() ![]() ![]() ![]() |
Related polyhedra and tiling
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
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
{∞,∞} | t{∞,∞} | r{∞,∞} | 2t{∞,∞}=t{∞,∞} | 2r{∞,∞}={∞,∞} | rr{∞,∞} | tr{∞,∞} |
Dual tilings | ||||||
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
V∞∞ | V∞.∞.∞ | V(∞.∞)2 | V∞.∞.∞ | V∞∞ | V4.∞.4.∞ | V4.4.∞ |
Alternations | ||||||
[1+,∞,∞] (*∞∞2) | [∞+,∞] (∞*∞) | [∞,1+,∞] (*∞∞∞∞) | [∞,∞+] (∞*∞) | [∞,∞,1+] (*∞∞2) | [(∞,∞,2+)] (2*∞∞) | [∞,∞]+ (2∞∞) |
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | |
h{∞,∞} | s{∞,∞} | hr{∞,∞} | s{∞,∞} | h2{∞,∞} | hrr{∞,∞} | sr{∞,∞} |
Alternation duals | ||||||
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![]() | ![]() | ![]() | ![]() | |||
V(∞.∞)∞ | V(3.∞)3 | V(∞.4)4 | V(3.∞)3 | V∞∞ | V(4.∞.4)2 | V3.3.∞.3.∞ |
Paracompact uniform tilings in [(∞,∞,∞)] family
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
(∞,∞,∞) 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
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
Wikimedia Commons has media related to Infinite-order apeirogonal tiling.
References
- John H. 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.
External links
- Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
- Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
- Hyperbolic and Spherical Tiling Gallery
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch