Tetraoctagonal tiling

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Tetraoctagonal tiling
Tetraoctagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.8)2
Schläfli symbol r{8,4} or
rr{8,8}
rr(4,4,4)
t0,1,2,3(∞,4,∞,4)
Wythoff symbol 2 | 8 4
Coxeter diagram or
or

Symmetry group [8,4], (*842)
[8,8], (*882)
[(4,4,4)], (*444)
[(∞,4,∞,4)], (*4242)
Dual Order-8-4 quasiregular rhombic tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.

Constructions

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There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1+], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1+,8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1+,8,4,1+], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).

Four uniform constructions of 4.8.4.8
Name Tetra-octagonal tiling Rhombi-octaoctagonal tiling
Image        
Symmetry [8,4]
(*842)
     
[8,8] = [8,4,1+]
(*882)
      =    
[(4,4,4)] = [1+,8,4]
(*444)
      =     
[(∞,4,∞,4)] = [1+,8,4,1+]
(*4242)
      =       or    
Schläfli r{8,4} rr{8,8}
=r{8,4}1/2
r(4,4,4)
=r{4,8}1/2
t0,1,2,3(∞,4,∞,4)
=r{8,4}1/4
Coxeter             =           =            =       or    

Symmetry

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The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) 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
Dimensional family of quasiregular polyhedra and tilings: (8.n)2
Symmetry
*8n2
[n,8]
Hyperbolic... Paracompact Noncompact
*832
[3,8]
*842
[4,8]
*852
[5,8]
*862
[6,8]
*872
[7,8]
*882
[8,8]...
*∞82
[∞,8]
 
[iπ/λ,8]
Coxeter                                                
Quasiregular
figures
configuration
 
3.8.3.8
 
4.8.4.8
 
8.5.8.5
 
8.6.8.6
 
8.7.8.7
 
8.8.8.8
 
8.∞.8.∞
 
8.∞.8.∞
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
Uniform (4,4,4) tilings
Symmetry: [(4,4,4)], (*444) [(4,4,4)]+
(444)
[(1+,4,4,4)]
(*4242)
[(4+,4,4)]
(4*22)
    
     
    
     
    
     
    
     
    
     
    
     
    
     
    
     
    
     
    
     
                   
t0(4,4,4)
h{8,4}
t0,1(4,4,4)
h2{8,4}
t1(4,4,4)
{4,8}1/2
t1,2(4,4,4)
h2{8,4}
t2(4,4,4)
h{8,4}
t0,2(4,4,4)
r{4,8}1/2
t0,1,2(4,4,4)
t{4,8}1/2
s(4,4,4)
s{4,8}1/2
h(4,4,4)
h{4,8}1/2
hr(4,4,4)
hr{4,8}1/2
Uniform duals
                   
V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V(4.4)4 V4.8.4.8 V8.8.8 V3.4.3.4.3.4 V88 V(4,4)3

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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