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Order-4 apeirogonal tiling
Regular tiling in geometry From Wikipedia, the free encyclopedia
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In geometry, the order-4 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,4}.
![]() | This article may be too technical for most readers to understand. (July 2013) |
Order-4 apeirogonal tiling | |
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![]() Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic regular tiling |
Vertex configuration | ∞4 |
Schläfli symbol | {∞,4} r{∞,∞} t(∞,∞,∞) t0,1,2,3(∞,∞,∞,∞) |
Wythoff symbol | 4 | ∞ 2 2 | ∞ ∞ ∞ ∞ | ∞ |
Coxeter diagram | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [∞,4], (*∞42) [∞,∞], (*∞∞2) [(∞,∞,∞)], (*∞∞∞) (*∞∞∞∞) |
Dual | Infinite-order square tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive edge-transitive |
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Symmetry
This tiling represents the mirror lines of *2∞ symmetry. Its dual tiling represents the fundamental domains of orbifold notation *∞∞∞∞ symmetry, a square domain with four ideal vertices.
Uniform colorings
Like the Euclidean square tiling there are 9 uniform colorings for this tiling, with 3 uniform colorings generated by triangle reflective domains. A fourth can be constructed from an infinite square symmetry (*∞∞∞∞) with 4 colors around a vertex. The checker board, r{∞,∞}, coloring defines the fundamental domains of [(∞,4,4)], (*∞44) symmetry, usually shown as black and white domains of reflective orientations.
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Related polyhedra and tiling
Summarize
Perspective
This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with four faces per vertex, starting with the octahedron, with Schläfli symbol {n,4}, and Coxeter diagram , with n progressing to infinity.
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See also
Wikimedia Commons has media related to Order-4 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
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