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Tetrahexagonal tiling

December 07, 2023

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 edit

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}1⁄2 r{6,4}1⁄2 r{6,4}1⁄4
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 edit

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.

    

Related polyhedra and tiling edit

*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 edit

 
Wikimedia Commons has media related to Uniform tiling 4-6-4-6.

References edit

  • 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.

External links edit

  • 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

December 07, 2023
tetrahexagonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, 2schläfli, symbol, displaystyle, begin, bmatrix, bmatrix, wythoff, symbol, 4coxeter, diagram, orsymmetry, group, 3232, dual, order, quasireg. Tetrahexagonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 4 6 2Schlafli symbol r 6 4 or 6 4 displaystyle begin Bmatrix 6 4 end Bmatrix rr 6 6 r 4 4 3 t0 1 2 3 3 3 Wythoff symbol 2 6 4Coxeter diagram or orSymmetry group 6 4 642 6 6 662 4 4 3 443 3 3 3232 Dual Order 6 4 quasiregular rhombic tilingProperties Vertex transitive edge transitiveIn geometry the tetrahexagonal tiling is a uniform tiling of the hyperbolic plane It has Schlafli symbol r 6 4 Contents 1 Constructions 2 Symmetry 3 Related polyhedra and tiling 4 See also 5 References 6 External linksConstructions editThere 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 UniformColoring nbsp nbsp nbsp nbsp FundamentalDomains nbsp nbsp nbsp nbsp Schlafli r 6 4 r 4 6 1 2 r 6 4 1 2 r 6 4 1 4Symmetry 6 4 642 nbsp nbsp nbsp nbsp nbsp 6 6 6 4 1 662 nbsp nbsp nbsp 4 4 3 1 6 4 443 nbsp nbsp nbsp 3 3 1 6 4 1 3232 nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp Symbol r 6 4 rr 6 6 r 4 3 4 t0 1 2 3 3 3 Coxeterdiagram nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp or nbsp nbsp nbsp Symmetry editThe 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 nbsp nbsp nbsp nbsp Related polyhedra and tiling edit n42 symmetry mutations of quasiregular tilings 4 n 2 vteSymmetry 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp Config 4 3 2 4 4 2 4 5 2 4 6 2 4 7 2 4 8 2 4 2 4 ni 2Symmetry mutation of quasiregular tilings 6 n 6 n vteSymmetry 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 ip l 6 Quasiregularfiguresconfiguration nbsp 6 3 6 3 nbsp 6 4 6 4 nbsp 6 5 6 5 nbsp 6 6 6 6 nbsp 6 7 6 7 nbsp 6 8 6 8 nbsp 6 6 6 6 Dual figuresRhombicfiguresconfiguration nbsp V6 3 6 3 nbsp V6 4 6 4 nbsp V6 5 6 5 nbsp V6 6 6 6 V6 7 6 7 nbsp V6 8 6 8 nbsp V6 6 Uniform tetrahexagonal tilings vteSymmetry 6 4 642 with 6 6 662 4 3 3 443 3 3222 index 2 subsymmetries And 3 3 3232 index 4 subsymmetry nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 6 4 t 6 4 r 6 4 t 4 6 4 6 rr 6 4 tr 6 4 Uniform duals nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V64 V4 12 12 V 4 6 2 V6 8 8 V46 V4 4 4 6 V4 8 12Alternations 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp h 6 4 s 6 4 hr 6 4 s 4 6 h 4 6 hrr 6 4 sr 6 4 Uniform hexahexagonal tilings vteSymmetry 6 6 662 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V66 V6 12 12 V6 6 6 6 V6 12 12 V66 V4 6 4 6 V4 12 12Alternations 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 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 vteSymmetry 4 4 3 443 4 4 3 443 4 4 3 3 22 4 1 4 3 3232 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp h 6 4 t0 4 4 3 h2 6 4 t0 1 4 4 3 4 6 1 2t1 4 4 3 h2 6 4 t1 2 4 4 3 h 6 4 t2 4 4 3 r 6 4 1 2t0 2 4 4 3 t 4 6 1 2t0 1 2 4 4 3 s 4 6 1 2s 4 4 3 hr 4 6 1 2hr 4 3 4 h 4 6 1 2h 4 3 4 q 4 6 h1 4 3 4 Uniform duals nbsp nbsp nbsp nbsp 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 6Similar H2 tilings in 3232 symmetry vteCoxeterdiagrams nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Vertexfigure 66 3 4 3 4 2 3 4 6 6 4 6 4 6 4Image nbsp nbsp nbsp nbsp Dual nbsp nbsp See also edit nbsp Wikimedia Commons has media related to Uniform tiling 4 6 4 6 Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopesReferences editJohn 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 External links editWeisstein Eric W Hyperbolic tiling MathWorld Weisstein Eric W Poincare hyperbolic disk MathWorld Hyperbolic and Spherical Tiling Gallery KaleidoTile 3 Educational software to create spherical planar and hyperbolic tilings Hyperbolic Planar Tessellations Don Hatch Retrieved from https en wikipedia org w index php title Tetrahexagonal tiling amp oldid 788527978, wikipedia, wiki, book, books, library,

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