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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 is a way of covering the hyperbolic plane—a non-Euclidean surface with constant negative curvature—with a repeating pattern of identical shapes without gaps or overlaps.
![]() | 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 |
This tiling is constructed from apeirogons, which are polygons with an infinite number of sides. In this specific pattern, four of these apeirogons meet at each vertex. It can be seen as the hyperbolic equivalent of the familiar square tiling of the Euclidean plane, where four squares meet at each vertex. Its Schläfli symbol is {∞,4}, with the "4" indicating that four polygons meet at a vertex and the "∞" indicating that the polygon used is an apeirogon.
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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
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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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