Infinite-order apeirogonal tiling
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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
[edit]This tiling represents the fundamental domains of *∞∞ symmetry.
Uniform colorings
[edit]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
[edit]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.
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 | ||||||
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V∞∞ | V∞.∞.∞.∞ | V∞∞ | V∞.∞.∞.∞ | V∞∞ | V∞.∞.∞.∞ | V∞.∞.∞ |
Alternations | ||||||
[(1+,∞,∞,∞)] (*∞∞∞∞) |
[∞+,∞,∞)] (∞*∞) |
[∞,1+,∞,∞)] (*∞∞∞∞) |
[∞,∞+,∞)] (∞*∞) |
[(∞,∞,∞,1+)] (*∞∞∞∞) |
[(∞,∞,∞+)] (∞*∞) |
[∞,∞,∞)]+ (∞∞∞) |
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Alternation duals | ||||||
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V(∞.∞)∞ | V(∞.4)4 | V(∞.∞)∞ | V(∞.4)4 | V(∞.∞)∞ | V(∞.4)4 | V3.∞.3.∞.3.∞ |
See also
[edit]![](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
[edit]- 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.