Infinite-order pentagonal tiling

Infinite-order pentagonal tiling
Infinite-order pentagonal 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 Order-5 apeirogonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In 2-dimensional hyperbolic geometry, the infinite-order pentagonal tiling is a regular tiling. It has Schläfli symbol of {5,∞}. All vertices are ideal, located at "infinity", seen on the boundary of the Poincaré hyperbolic disk projection.

Symmetry

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There is a half symmetry form,     , seen with alternating colors:

 
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This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (5n).

Finite Compact hyperbolic Paracompact
 
{5,3}
     
 
{5,4}
     
 
{5,5}
     
 
{5,6}
     
 
{5,7}
     
 
{5,8}...
     
 
{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 (2008). "Chapter 19, The Hyperbolic Archimedean Tessellations". The Symmetries of Things. ISBN 978-1-56881-220-5.
  • H. S. M. Coxeter (1999). "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. ISBN 0-486-40919-8. LCCN 99035678.
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