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 CDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel 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[]

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

H2chess 26ib.png

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

Related polyhedra and tiling[]

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,6}, and Coxeter diagram CDel node 1.pngCDel n.pngCDel node.pngCDel 6.pngCDel node.png, with n progressing to infinity.

Regular tilings {n,6}
  • v
Spherical Euclidean Hyperbolic tilings
Spherical hexagonal hosohedron.png
{2,6}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 6.pngCDel node.png 		Uniform tiling 63-t2.svg
{3,6}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 6.pngCDel node.png 		H2 tiling 246-4.png
{4,6}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 6.pngCDel node.png 		H2 tiling 256-4.png
{5,6}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 6.pngCDel node.png 		H2 tiling 266-4.png
{6,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png 		H2 tiling 267-1.png

CDel node 1.pngCDel 7.pngCDel node.pngCDel 6.pngCDel node.png 		H2 tiling 268-1.png
{8,6}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 6.pngCDel node.png 		... H2 tiling 26i-1.png
{∞,6}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 6.pngCDel node.png

See also[]

  • Tilings of regular polygons
  • List of uniform planar tilings
  • List of regular polytopes

References[]

  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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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