Order-5 square tiling

Summary

Order-5 square tiling
Order-5 square tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 45
Schläfli symbol {4,5}
Wythoff symbol 5 | 4 2
Coxeter diagram CDel node.pngCDel 5.pngCDel node.pngCDel 4.pngCDel node 1.png
Symmetry group [5,4], (*542)
Dual Order-4 pentagonal tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {4,5}.

Related polyhedra and tilingEdit

Spherical Hyperbolic tilings
 
{2,5}
     
 
{3,5}
     
 
{4,5}       
{5,5}
     
 
{6,5}
     
 
{7,5}
     
 
{8,5}
     
...  
{∞,5}
     

This tiling is topologically related as a part of sequence of regular polyhedra and tilings with vertex figure (4n).

*n42 symmetry mutation of regular tilings: {4,n}
Spherical Euclidean Compact hyperbolic Paracompact
 
{4,3}
     
 
{4,4}
     
 
{4,5}       
{4,6}
     
 
{4,7}
     
 
{4,8}...
     
 
{4,∞}
     
Uniform pentagonal/square tilings
Symmetry: [5,4], (*542) [5,4]+, (542) [5+,4], (5*2) [5,4,1+], (*552)
                                                           
                   
{5,4} t{5,4} r{5,4} 2t{5,4}=t{4,5} 2r{5,4}={4,5} rr{5,4} tr{5,4} sr{5,4} s{5,4} h{4,5}
Uniform duals
                                                           
                 
V54 V4.10.10 V4.5.4.5 V5.8.8 V45 V4.4.5.4 V4.8.10 V3.3.4.3.5 V3.3.5.3.5 V55

This hyperbolic tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space.

 

ReferencesEdit

  • 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.

See alsoEdit

External linksEdit

  • Weisstein, Eric W. "Hyperbolic tiling". MathWorld.
  • Weisstein, Eric W. "Poincaré hyperbolic disk". MathWorld.
  • Hyperbolic and Spherical Tiling Gallery
  • KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
  • Hyperbolic Planar Tessellations, Don Hatch