Rhombitetrahexagonal tiling

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Rhombitetrahexagonal tiling
Rhombitetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 4.4.6.4
Schläfli symbol rr{6,4} or
Wythoff symbol 4 | 6 2
Coxeter diagram

Symmetry group [6,4], (*642)
Dual Deltoidal tetrahexagonal tiling
Properties Vertex-transitive

In geometry, the rhombitetrahexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{6,4}. It can be seen as constructed as a rectified tetrahexagonal tiling, r{6,4}, as well as an expanded order-4 hexagonal tiling or expanded order-6 square tiling.

Constructions

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There are two uniform constructions of this tiling, one from [6,4] or (*642) symmetry, and secondly removing the mirror middle, [6,1+,4], gives a rectangular fundamental domain [∞,3,∞], (*3222).

Two uniform constructions of 4.4.4.6
Name Rhombitetrahexagonal tiling
Image    
Symmetry [6,4]
(*642)
     
[6,1+,4] = [∞,3,∞]
(*3222)
      =    
Schläfli symbol rr{6,4} t0,1,2,3{∞,3,∞}
Coxeter diagram             =    

There are 3 lower symmetry forms seen by including edge-colorings:       sees the hexagons as truncated triangles, with two color edges, with [6,4+] (4*3) symmetry.       sees the yellow squares as rectangles, with two color edges, with [6+,4] (6*2) symmetry. A final quarter symmetry combines these colorings, with [6+,4+] (32×) symmetry, with 2 and 3 fold gyration points and glide reflections.

This four color tiling is related to a semiregular infinite skew polyhedron with the same vertex figure in Euclidean 3-space with a prismatic honeycomb construction of        .

 

Symmetry

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The dual tiling, called a deltoidal tetrahexagonal tiling, represents the fundamental domains of the *3222 orbifold, shown here from three different centers. Its fundamental domain is a Lambert quadrilateral, with 3 right angles. This symmetry can be seen from a [6,4], (*642) triangular symmetry with one mirror removed, constructed as [6,1+,4], (*3222). Removing half of the blue mirrors doubles the domain again into *3322 symmetry.

    
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*n42 symmetry mutation of expanded tilings: n.4.4.4
Symmetry
[n,4], (*n42)
Spherical Euclidean Compact hyperbolic Paracomp.
*342
[3,4]
*442
[4,4]
*542
[5,4]
*642
[6,4]
*742
[7,4]
*842
[8,4]
*∞42
[∞,4]
Expanded
figures
             
Config. 3.4.4.4 4.4.4.4 5.4.4.4 6.4.4.4 7.4.4.4 8.4.4.4 ∞.4.4.4
Rhombic
figures
config.
 
V3.4.4.4
 
V4.4.4.4
 
V5.4.4.4
 
V6.4.4.4
 
V7.4.4.4
 
V8.4.4.4
 
V∞.4.4.4
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 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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