Order-4 hexagonal tiling

Summary

Order-4 hexagonal tiling
Order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex configuration 64
Schläfli symbol {6,4}
Wythoff symbol 4 | 6 2
Coxeter diagram
Symmetry group [6,4], (*642)
Dual Order-6 square tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the order-4 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,4}.

Symmetry

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This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *222222 with 6 order-2 mirror intersections. In Coxeter notation can be represented as [6*,4], removing two of three mirrors (passing through the hexagon center). Adding a bisecting mirror through 2 vertices of a hexagonal fundamental domain defines a trapezohedral *4422 symmetry. Adding 3 bisecting mirrors through the vertices defines *443 symmetry. Adding 3 bisecting mirrors through the edge defines *3222 symmetry. Adding all 6 bisectors leads to full *642 symmetry.

 
*222222
 
*443
 
*3222
 
*642

Uniform colorings

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There are 7 distinct uniform colorings for the order-4 hexagonal tiling. They are similar to 7 of the uniform colorings of the square tiling, but exclude 2 cases with order-2 gyrational symmetry. Four of them have reflective constructions and Coxeter diagrams while three of them are undercolorings.

Uniform constructions of 6.6.6.6
1 color 2 colors 3 and 2 colors 4, 3 and 2 colors
Uniform
Coloring
 
(1111)
 
(1212)
 
(1213)
 
(1113)
 
(1234)
 
(1123)
 
(1122)
Symmetry [6,4]
(*642)
     
[6,6]
(*662)
    =      
[(6,6,3)] = [6,6,1+]
(*663)
    =      
[1+,6,6,1+]
(*3333)
    =       =      
Symbol {6,4} r{6,6} = {6,4}1/2 r(6,3,6) = r{6,6}1/2 r{6,6}1/4
Coxeter
diagram
          =           =           =       =      

Regular maps

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The regular map {6,4}3 or {6,4}(4,0) can be seen as a 4-coloring on the {6,4} tiling. It also has a representation as a petrial octahedron, {3,4}π, an abstract polyhedron with vertices and edges of an octahedron, but instead connected by 4 Petrie polygon faces.

  
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This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram      , progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n}
Spherical Euclidean Hyperbolic tilings
 
{6,2}
 
{6,3}
 
{6,4}  
{6,5}
 
{6,6}
 
{6,7}
 
{6,8}
...  
{6,∞}

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.

*n42 symmetry mutation of regular tilings: {n,4}
Spherical Euclidean Hyperbolic tilings
               
24 34 44 54 64 74 84 ...4
Symmetry mutation of quasiregular tilings: (6.n)2
Symmetry
*6n2
[n,6]
Euclidean Compact hyperbolic Paracompact Noncompact
*632
[3,6]
*642
[4,6]
*652
[5,6]
*662
[6,6]
*762
[7,6]
*862
[8,6]...
*∞62
[∞,6]
 
[iπ/λ,6]
Quasiregular
figures
configuration
 
6.3.6.3
 
6.4.6.4
 
6.5.6.5
 
6.6.6.6
 
6.7.6.7
 
6.8.6.8
 
6.∞.6.∞

6.∞.6.∞
Dual figures
Rhombic
figures
configuration
 
V6.3.6.3
 
V6.4.6.4  
V6.5.6.5
 
V6.6.6.6

V6.7.6.7
 
V6.8.6.8
 
V6.∞.6.∞
Uniform tetrahexagonal tilings
Symmetry: [6,4], (*642)
(with [6,6] (*662), [(4,3,3)] (*443) , [∞,3,∞] (*3222) index 2 subsymmetries)
(And [(∞,3,∞,3)] (*3232) index 4 subsymmetry)
     
=    
 
=    
=    
     
=    
     
=    
=    
 
=    
     
 
=    
     
 
=    
=    
=      
     
 
 
=    
     
             
{6,4} t{6,4} r{6,4} t{4,6} {4,6} rr{6,4} tr{6,4}
Uniform duals
                                         
             
V64 V4.12.12 V(4.6)2 V6.8.8 V46 V4.4.4.6 V4.8.12
Alternations
[1+,6,4]
(*443)
[6+,4]
(6*2)
[6,1+,4]
(*3222)
[6,4+]
(4*3)
[6,4,1+]
(*662)
[(6,4,2+)]
(2*32)
[6,4]+
(642)
     
=    
     
=     
     
=    
     
=    
     
=    
     
=     
     
             
h{6,4} s{6,4} hr{6,4} s{4,6} h{4,6} hrr{6,4} sr{6,4}
Uniform hexahexagonal tilings
Symmetry: [6,6], (*662)
      =    
=      
      =    
=      
      =    
=      
      =    
=      
      =    
=      
      =    
=      
      =   
=      
             
{6,6}
= h{4,6}
t{6,6}
= h2{4,6}
r{6,6}
{6,4}
t{6,6}
= h2{4,6}
{6,6}
= h{4,6}
rr{6,6}
r{6,4}
tr{6,6}
t{6,4}
Uniform duals
                                         
             
V66 V6.12.12 V6.6.6.6 V6.12.12 V66 V4.6.4.6 V4.12.12
Alternations
[1+,6,6]
(*663)
[6+,6]
(6*3)
[6,1+,6]
(*3232)
[6,6+]
(6*3)
[6,6,1+]
(*663)
[(6,6,2+)]
(2*33)
[6,6]+
(662)
      =                 =                 =                
                                         
         
h{6,6} s{6,6} hr{6,6} s{6,6} h{6,6} hrr{6,6} sr{6,6}
Similar H2 tilings in *3232 symmetry
Coxeter
diagrams
                       
                                   
               
Vertex
figure
66 (3.4.3.4)2 3.4.6.6.4 6.4.6.4
Image        
Dual    
Uniform tilings in symmetry *3222
    64      6.6.4.4
 
    (3.4.4)2
 
    4.3.4.3.3.3
 
    6.6.4.4
 
    6.4.4.4
 
    3.4.4.4.4
 
    (3.4.4)2
 
    3.4.4.4.4
 
    46
 

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