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Alternated order-4 hexagonal tiling

Alternated order-4 hexagonal tiling
Alternated order-4 hexagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (3.4)4
Schläfli symbol h{6,4} or (3,4,4)
Wythoff symbol 4 | 3 4
Coxeter diagram or
Symmetry group [(4,4,3)], (*443)
Dual Order-4-4-3_t0 dual tiling
Properties Vertex-transitive

In geometry, the alternated order-4 hexagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of (3,4,4), h{6,4}, and hr{6,6}.

Uniform constructions

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There are four uniform constructions, with some of lower ones which can be seen with two colors of triangles:

*443 3333 *3232 3*22
      =           =           =     =           =    
   
(4,4,3) = h{6,4} hr{6,6} = h{6,4}1⁄2
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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    

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.

See also

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Wikimedia Commons has media related to Uniform tiling 3-4-3-4-3-4-3-4.
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