Order-5 apeirogonal tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | ∞5 |
Schläfli symbol | {∞,5} |
Wythoff symbol | 5 | ∞ 2 |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [∞,5], (*∞52) |
Dual | Infinite-order pentagonal tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive edge-transitive |
In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.
Symmetry
[edit]The dual to this tiling represents the fundamental domains of [∞,5*] symmetry, orbifold notation *∞∞∞∞∞ symmetry, a pentagonal domain with five ideal vertices.
The order-5 apeirogonal tiling can be uniformly colored with 5 colored apeirogons around each vertex, and coxeter diagram: , except ultraparallel branches on the diagonals.
Related polyhedra and tiling
[edit]This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with five faces per vertex, starting with the icosahedron, with Schläfli symbol {n,5}, and Coxeter diagram , with n progressing to infinity.
Spherical | Hyperbolic tilings | |||||||
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![]() {2,5} ![]() ![]() ![]() ![]() ![]() |
![]() {3,5} ![]() ![]() ![]() ![]() ![]() |
![]() {4,5} ![]() ![]() ![]() ![]() ![]() |
![]() {5,5} ![]() ![]() ![]() ![]() ![]() |
![]() {6,5} ![]() ![]() ![]() ![]() ![]() |
![]() {7,5} ![]() ![]() ![]() ![]() ![]() |
![]() {8,5} ![]() ![]() ![]() ![]() ![]() |
... | ![]() {∞,5} ![]() ![]() ![]() ![]() ![]() |
Paracompact uniform apeirogonal/pentagonal tilings | |||||||||||
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Symmetry: [∞,5], (*∞52) | [∞,5]+ (∞52) |
[1+,∞,5] (*∞55) |
[∞,5+] (5*∞) | ||||||||
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{∞,5} | t{∞,5} | r{∞,5} | 2t{∞,5}=t{5,∞} | 2r{∞,5}={5,∞} | rr{∞,5} | tr{∞,5} | sr{∞,5} | h{∞,5} | h2{∞,5} | s{5,∞} | |
Uniform duals | |||||||||||
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V∞5 | V5.∞.∞ | V5.∞.5.∞ | V∞.10.10 | V5∞ | V4.5.4.∞ | V4.10.∞ | V3.3.5.3.∞ | V(∞.5)5 | V3.5.3.5.3.∞ |
See also
[edit]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.