Order-5 apeirogonal tiling

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
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

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

 

The order-5 apeirogonal tiling can be uniformly colored with 5 colored apeirogons around each vertex, and coxeter diagram:       , 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 five faces per vertex, starting with the icosahedron, with Schläfli symbol {n,5}, and Coxeter diagram      , with n progressing to infinity.

Spherical Hyperbolic tilings
 
{2,5}
     
 
{3,5}
     
 
{4,5}
     
 
{5,5}
     
 
{6,5}
     
 
{7,5}
     
 
{8,5}
     
...  
{∞,5}
     
Paracompact uniform apeirogonal/pentagonal tilings
Symmetry: [∞,5], (*∞52) [∞,5]+
(∞52)
[1+,∞,5]
(*∞55)
[∞,5+]
(5*∞)
                                                                 
                 
{∞,5} t{∞,5} r{∞,5} 2t{∞,5}=t{5,∞} 2r{∞,5}={5,∞} rr{∞,5} tr{∞,5} sr{∞,5} h{∞,5} h2{∞,5} s{5,∞}
Uniform duals
                                                                 
       
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

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References

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  • 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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