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[edit]

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[edit]

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
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 tiling 245-4.png
{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
{7,5}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 258-1.png
{8,5}
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

See also[edit]

References[edit]

  • 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[edit]