Order-8 octagonal tiling

Summary

Order-8 octagonal tiling
Order-8 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 88
Schläfli symbol {8,8}
Wythoff symbol 8 | 8 2
Coxeter diagram
Symmetry group [8,8], (*882)
Dual self dual
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-8 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,8} (eight octagons around each vertex) and is self-dual.

Symmetry

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This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting at a point and bounding regular octagon fundamental domains. This symmetry by orbifold notation is called *44444444 with 8 order-4 mirror intersections. In Coxeter notation can be represented as [8,8*], removing two of three mirrors (passing through the octagon center) in the [8,8] symmetry.

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This tiling is topologically related as a part of sequence of regular tilings with octagonal faces, starting with the octagonal tiling, with Schläfli symbol {8,n}, and Coxeter diagram      , progressing to infinity.

n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling              
Config. 8.8 83 84 85 86 87 88 ...8
Regular tilings: {n,8}
Spherical Hyperbolic tilings
 
{2,8}
     
 
{3,8}
     
 
{4,8}
     
 
{5,8}
     
 
{6,8}
     
 
{7,8}
     
 
{8,8}      ...  
{∞,8}
     
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
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{8,8} t{8,8}
r{8,8} 2t{8,8}=t{8,8} 2r{8,8}={8,8} rr{8,8} tr{8,8}
Uniform duals
                                         
             
V88 V8.16.16 V8.8.8.8 V8.16.16 V88 V4.8.4.8 V4.16.16 Alternations [1+,8,8](*884) [8+,8](8*4) [8,1+,8](*4242) [8,8+](8*4) [8,8,1+](*884) [(8,8,2+)](2*44) [8,8]+(882)       =                  =                 =           =    =             =    =                
h{8,8} s{8,8} hr{8,8} s{8,8} h{8,8} hrr{8,8} sr{8,8}
Alternation duals
                                         
   
V(4.8)8 V3.4.3.8.3.8 V(4.4)4 V3.4.3.8.3.8 V(4.8)8 V46 V3.3.8.3.8

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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  • 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