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Order-4 apeirogonal tiling

Regular tiling in geometry From Wikipedia, the free encyclopedia

Order-4 apeirogonal tiling
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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}.

Order-4 apeirogonal tiling
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Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration4
Schläfli symbol{,4}
r{,}
t(,,)
t0,1,2,3(∞,∞,∞,∞)
Wythoff symbol4 | 2
2 |
|
Coxeter diagram

Symmetry group[,4], (*42)
[,], (*2)
[(,,)], (*)
(*)
DualInfinite-order square tiling
PropertiesVertex-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.

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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.

More information 1 color, 2 color ...
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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.

More information Spherical, Euclidean ...
More information Dual figures, Alternations ...
More information Dual tilings, Alternations ...
More information Dual tilings, Alternations ...
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See also

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.
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