Order-4 octagonal tiling

Summary

Order-4 octagonal tiling
Order-4 octagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 84
Schläfli symbol {8,4}
r{8,8}
Wythoff symbol 4 | 8 2
Coxeter diagram
or
Symmetry group [8,4], (*842)
[8,8], (*882)
Dual Order-8 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 octagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {8,4}. Its checkerboard coloring can be called a octaoctagonal tiling, and Schläfli symbol of r{8,8}.

Uniform constructions

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There are four uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,8] kaleidoscope. Removing the mirror between the order 2 and 4 points, [8,8,1+], gives [(8,8,4)], (*884) symmetry. Removing two mirrors as [8,4*], leaves remaining mirrors *4444 symmetry.

Four uniform constructions of 8.8.8.8
Uniform
Coloring
       
Symmetry [8,4]
(*842)
     
[8,8]
(*882)
      =      
[(8,4,8)] = [8,8,1+]
(*884)
      =     

      =     

[1+,8,8,1+]
(*4444)
      =
     
Symbol {8,4} r{8,8} r(8,4,8) = r{8,8}12 r{8,4}18 = r{8,8}14
Coxeter
diagram
                  =     

      =     

      =       =
      =     

Symmetry

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This tiling represents a hyperbolic kaleidoscope of 8 mirrors meeting as edges of a regular hexagon. This symmetry by orbifold notation is called (*22222222) or (*28) with 8 order-2 mirror intersections. In Coxeter notation can be represented as [8*,4], removing two of three mirrors (passing through the octagon center) in the [8,4] symmetry. Adding a bisecting mirror through 2 vertices of an octagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 4 bisecting mirrors through the vertices defines *444 symmetry. Adding 4 bisecting mirrors through the edge defines *4222 symmetry. Adding all 8 bisectors leads to full *842 symmetry.

 
*444
 
*4222
 
*832

The kaleidoscopic domains can be seen as bicolored octagonal tiling, representing mirror images of the fundamental domain. This coloring represents the uniform tiling r{8,8}, a quasiregular tiling and it can be called a octaoctagonal tiling.

   
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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.

*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
               
24 34 44 54 64 74 84 ...4
Regular tilings: {n,8}
Spherical Hyperbolic tilings
 
{2,8}
     
 
{3,8}
     
 
{4,8}
     
 
{5,8}
     
 
{6,8}
     
 
{7,8}
     
 
{8,8}
     
...  
{∞,8}
     

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,4}, and Coxeter diagram      , with n progressing to infinity.

 
{3,4}
     
 
{4,4}
     
 
{5,4}
     
 
{6,4}
     
 
{7,4}
     
 
{8,4}
     
...  
{∞,4}
     
Uniform octagonal/square tilings
[8,4], (*842)
(with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries)
(And [(∞,4,∞,4)] (*4242) index 4 subsymmetry)
     
=    
 
=     
=      
     
=    
     
=    
=     
 
=      
     
 
=     
     
 
=     
=     
     
 
 
=     
     
             
{8,4} t{8,4}
r{8,4} 2t{8,4}=t{4,8} 2r{8,4}={4,8} rr{8,4} tr{8,4}
Uniform duals
                                         
             
V84 V4.16.16 V(4.8)2 V8.8.8 V48 V4.4.4.8 V4.8.16
Alternations
[1+,8,4]
(*444)
[8+,4]
(8*2)
[8,1+,4]
(*4222)
[8,4+]
(4*4)
[8,4,1+]
(*882)
[(8,4,2+)]
(2*42)
[8,4]+
(842)
     
=     
     
=    
     
=     
     
=     
     
=    
     
=     
     
             
h{8,4} s{8,4} hr{8,4} s{4,8} h{4,8} hrr{8,4} sr{8,4}
Alternation duals
                                         
         
V(4.4)4 V3.(3.8)2 V(4.4.4)2 V(3.4)3 V88 V4.44 V3.3.4.3.8
Uniform octaoctagonal tilings
Symmetry: [8,8], (*882)
      =    
=      
      =    
=      
      =    
=      
      =    
=      
      =    
=      
      =    
=      
      =    
=      
             
{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