Order-5 apeirogonal tiling

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Order-5 apeirogonal tiling
Order-5 apeirogonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration ∞5
Schläfli symbol {∞,5}
Wythoff symbol 5 | ∞ 2
Coxeter diagram CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png
Symmetry group [∞,5], (*∞52)
Dual Infinite-order pentagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive edge-transitive

In geometry, the order-5 apeirogonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {∞,5}.

Symmetry[]

The dual to this tiling represents the fundamental domains of [∞,5*] symmetry, orbifold notation *∞∞∞∞∞ symmetry, a pentagonal domain with five ideal vertices.

H2chess 25ib.png

The order-5 apeirogonal tiling can be uniformly colored with 5 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,5}, and Coxeter diagram CDel node 1.pngCDel n.pngCDel node.pngCDel 5.pngCDel node.png, with n progressing to infinity.

Spherical Hyperbolic tilings
  • v
Spherical pentagonal hosohedron.png
{2,5}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 5.pngCDel node.png 		Uniform tiling 532-t2.png
{3,5}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 5.pngCDel node.png 		H2-5-4-primal.svg
{4,5}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 255-1.png
{5,5}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 256-1.png
{6,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 257-1.png

CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.png 		H2 tiling 258-1.png

CDel node 1.pngCDel 8.pngCDel node.pngCDel 5.pngCDel node.png 		... H2 tiling 25i-1.png
{∞,5}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png
Paracompact uniform apeirogonal/pentagonal tilings
  • v
Symmetry: [∞,5], (*∞52) [∞,5]+
(∞52) 		[1+,∞,5]
(*∞55) 		[∞,5+]
(5*∞)
CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 5.pngCDel node.png CDel node.pngCDel infin.pngCDel node 1.pngCDel 5.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node 1.png CDel node 1.pngCDel infin.pngCDel node 1.pngCDel 5.pngCDel node 1.png CDel node h.pngCDel infin.pngCDel node h.pngCDel 5.pngCDel node h.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node.png CDel node h1.pngCDel infin.pngCDel node.pngCDel 5.pngCDel node 1.png CDel node.pngCDel infin.pngCDel node h.pngCDel 5.pngCDel node h.png
H2 tiling 25i-1.png H2 tiling 25i-3.png H2 tiling 25i-2.png H2 tiling 25i-6.png H2 tiling 25i-4.png H2 tiling 25i-5.png H2 tiling 25i-7.png Uniform tiling i52-snub.png H2 tiling 55i-1.png
{∞,5} r{∞,5} 2r{∞,5}={5,∞} h{∞,5} h2{∞,5} s{5,∞}
Uniform duals
CDel node f1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.png CDel node.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node f1.png CDel node f1.pngCDel infin.pngCDel node f1.pngCDel infin.pngCDel node f1.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node 1.png CDel node fh.pngCDel infin.pngCDel node.pngCDel infin.pngCDel node.png CDel node.pngCDel infin.pngCDel node fh.pngCDel infin.pngCDel node fh.png
H2chess 25ib.png H2chess 25ie.png H2 tiling 25i-1.png H2checkers 25i.png
V∞5 V5.∞.∞ V5.∞.5.∞ V∞.10.10 V5∞ V4.5.4.∞ V4.10.∞ V3.3.5.3.∞ V(∞.5)5 V3.5.3.5.3.∞

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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