Snub tetrahexagonal tiling

Summary

Snub tetrahexagonal tiling
Snub tetrahexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration 3.3.4.3.6
Schläfli symbol sr{6,4} or
Wythoff symbol | 6 4 2
Coxeter diagram CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png or CDel node h.pngCDel split1-64.pngCDel nodes hh.png
Symmetry group [6,4]+, (642)
Dual Order-6-4 floret pentagonal tiling
Properties Vertex-transitive Chiral

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

ImagesEdit

Drawn in chiral pairs, with edges missing between black triangles:

  

Related polyhedra and tilingEdit

The snub tetrahexagonal tiling is fifth in a series of snub polyhedra and tilings with vertex figure 3.3.4.3.n.

4n2 symmetry mutations of snub tilings: 3.3.4.3.n
Symmetry
4n2
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Snub
figures
               
Config. 3.3.4.3.2 3.3.4.3.3 3.3.4.3.4 3.3.4.3.5 3.3.4.3.6 3.3.4.3.7 3.3.4.3.8 3.3.4.3.∞
Gyro
figures
       
Config. V3.3.4.3.2 V3.3.4.3.3 V3.3.4.3.4 V3.3.4.3.5 V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞
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} ReferencesEdit
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, 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.

See alsoEdit

External linksEdit

  • 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