Tetrahexagonal tiling

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Tetrahexagonal tiling
Tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.6)2
Schläfli symbol r{6,4} or
rr{6,6}
r(4,4,3)
t0,1,2,3(∞,3,∞,3)
Wythoff symbol 2 | 6 4
Coxeter diagram or
or

Symmetry group [6,4], (*642)
[6,6], (*662)
[(4,4,3)], (*443)
[(∞,3,∞,3)], (*3232)
Dual Order-6-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol r{6,4}.

Constructions

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There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [6,4] kaleidoscope. Removing the last mirror, [6,4,1+], gives [6,6], (*662). Removing the first mirror [1+,6,4], gives [(4,4,3)], (*443). Removing both mirror as [1+,6,4,1+], leaving [(3,∞,3,∞)] (*3232).

Four uniform constructions of 4.6.4.6
Uniform
Coloring
       
Fundamental
Domains
       
Schläfli r{6,4} r{4,6}12 r{6,4}12 r{6,4}14
Symmetry [6,4]
(*642)
     
[6,6] = [6,4,1+]
(*662)
   
[(4,4,3)] = [1+,6,4]
(*443)
   
[(∞,3,∞,3)] = [1+,6,4,1+]
(*3232)
      or    
Symbol r{6,4} rr{6,6} r(4,3,4) t0,1,2,3(∞,3,∞,3)
Coxeter
diagram
            =           =           =
      or    

Symmetry

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The dual tiling, called a rhombic tetrahexagonal tiling, with face configuration V4.6.4.6, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*3232), shown here in two different centered views. Adding a 2-fold rotation point in the center of each rhombi represents a (2*32) orbifold.

    
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*n42 symmetry mutations of quasiregular tilings: (4.n)2
Symmetry
*4n2
[n,4]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]...
*∞42
[∞,4]
 
[ni,4]
Figures              
Config. (4.3)2 (4.4)2 (4.5)2 (4.6)2 (4.7)2 (4.8)2 (4.∞)2 (4.ni)2
Symmetry mutation of quasiregular tilings: 6.n.6.n
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}
Uniform (4,4,3) tilings
Symmetry: [(4,4,3)] (*443) [(4,4,3)]+
(443)
[(4,4,3+)]
(3*22)
[(4,1+,4,3)]
(*3232)
                                           
                                                                 
                     
h{6,4}
t0(4,4,3)
h2{6,4}
t0,1(4,4,3)
{4,6}1/2
t1(4,4,3)
h2{6,4}
t1,2(4,4,3)
h{6,4}
t2(4,4,3)
r{6,4}1/2
t0,2(4,4,3)
t{4,6}1/2
t0,1,2(4,4,3)
s{4,6}1/2
s(4,4,3)
hr{4,6}1/2
hr(4,3,4)
h{4,6}1/2
h(4,3,4)
q{4,6}
h1(4,3,4)
Uniform duals
       
V(3.4)4 V3.8.4.8 V(4.4)3 V3.8.4.8 V(3.4)4 V4.6.4.6 V6.8.8 V3.3.3.4.3.4 V(4.4.3)2 V66 V4.3.4.6.6
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    

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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