Order-6 apeirogonal tiling

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Order-6 apeirogonal tiling
Order-6 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration ∞6
Schläfli symbol {∞,6}
Wythoff symbol 6 | ∞ 2
Coxeter diagram File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png
Symmetry group [∞,6], (*∞62)
Dual Infinite-order hexagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive edge-transitive

In geometry, the order-6 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,6}.

Symmetry

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The dual to this tiling represents the fundamental domains of [∞,6*] symmetry, orbifold notation *∞∞∞∞∞∞ symmetry, a hexagonal domain with five ideal vertices.

File:H2chess 26ib.png

The order-6 apeirogonal tiling can be uniformly colored with 6 colored apeirogons around each vertex, and coxeter diagram: File:CDel labelinfin.pngFile:CDel branch 11.pngFile:CDel iaib.pngFile:CDel nodes 11.pngFile:CDel split2-ii.pngFile:CDel node 1.png, except ultraparallel branches on the diagonals.

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This tiling is also topologically related as a part of sequence of regular polyhedra and tilings with six faces per vertex, starting with the triangular tiling, with Schläfli symbol {n,6}, and Coxeter diagram File:CDel node 1.pngFile:CDel n.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png, with n progressing to infinity.

Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings
File:Spherical hexagonal hosohedron.svg
{2,6}
File:CDel node 1.pngFile:CDel 2.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png 		File:Uniform tiling 63-t2.svg
{3,6}
File:CDel node 1.pngFile:CDel 3.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png 		File:H2 tiling 246-4.png
{4,6}
File:CDel node 1.pngFile:CDel 4.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png 		File:H2 tiling 256-4.png
{5,6}
File:CDel node 1.pngFile:CDel 5.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png 		File:H2 tiling 266-4.png
{6,6}
File:CDel node 1.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png 		File:H2 tiling 267-1.png
{7,6}
File:CDel node 1.pngFile:CDel 7.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png 		File:H2 tiling 268-1.png
{8,6}
File:CDel node 1.pngFile:CDel 8.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png 		... File:H2 tiling 26i-1.png
{∞,6}
File:CDel node 1.pngFile:CDel infin.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png

See also

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Wikimedia Commons has media related to Order-6 apeirogonal tiling.

References

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  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapter 19, The Hyperbolic Archimedean Tessellations)
  • Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
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