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Order-6 hexagonal tiling

December 15, 2023

In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.

Symmetry edit

This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *333333 with 6 order-3 mirror intersections. In Coxeter notation can be represented as [6*,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.

The even/odd fundamental domains of this kaleidoscope can be seen in the alternating colorings of the     tiling:

 

Related polyhedra and tiling edit

This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram      , progressing to infinity.

Regular tilings {n,6}
Spherical Euclidean Hyperbolic tilings
 
{2,6}
      		 
{3,6}
      		 
{4,6}
      		 
{5,6}
      		 
{6,6}
      		 
{7,6}
      		 
{8,6}
      		...  
{∞,6}
     

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram      , progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n}
Spherical Euclidean Hyperbolic tilings
 
{6,2} 		 
{6,3} 		 
{6,4} 		 
{6,5} 		 
{6,6} 		 
{6,7} 		 
{6,8} 		...  
{6,∞}
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}
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 edit

  • 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 edit

 
Wikimedia Commons has media related to Order-6 hexagonal tiling.

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 15, 2023
order, hexagonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, regular, tilingvertex, configuration, 66schläfli, symbol, wythoff, symbol, 2coxeter, diagramsymmetry, group, dual, self, dualproperties, vertex, transitive, edge, transitive, f. Order 6 hexagonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic regular tilingVertex configuration 66Schlafli symbol 6 6 Wythoff symbol 6 6 2Coxeter diagramSymmetry group 6 6 662 Dual self dualProperties Vertex transitive edge transitive face transitiveIn geometry the order 6 hexagonal tiling is a regular tiling of the hyperbolic plane It has Schlafli symbol of 6 6 and is self dual Contents 1 Symmetry 2 Related polyhedra and tiling 3 References 4 See also 5 External linksSymmetry editThis tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain This symmetry by orbifold notation is called 333333 with 6 order 3 mirror intersections In Coxeter notation can be represented as 6 6 removing two of three mirrors passing through the hexagon center in the 6 6 symmetry The even odd fundamental domains of this kaleidoscope can be seen in the alternating colorings of the nbsp nbsp nbsp tiling nbsp Related polyhedra and tiling editThis tiling is topologically related as a part of sequence of regular tilings with order 6 vertices with Schlafli symbol n 6 and Coxeter diagram nbsp nbsp nbsp nbsp nbsp progressing to infinity Regular tilings n 6 vteSpherical Euclidean Hyperbolic tilings nbsp 2 6 nbsp nbsp nbsp nbsp nbsp nbsp 3 6 nbsp nbsp nbsp nbsp nbsp nbsp 4 6 nbsp nbsp nbsp nbsp nbsp nbsp 5 6 nbsp nbsp nbsp nbsp nbsp nbsp 6 6 nbsp nbsp nbsp nbsp nbsp nbsp 7 6 nbsp nbsp nbsp nbsp nbsp nbsp 8 6 nbsp nbsp nbsp nbsp nbsp nbsp 6 nbsp nbsp nbsp nbsp nbsp This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces starting with the hexagonal tiling with Schlafli symbol 6 n and Coxeter diagram nbsp nbsp nbsp nbsp nbsp progressing to infinity n62 symmetry mutation of regular tilings 6 n vteSpherical Euclidean Hyperbolic tilings nbsp 6 2 nbsp 6 3 nbsp 6 4 nbsp 6 5 nbsp 6 6 nbsp 6 7 nbsp 6 8 nbsp 6 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 Similar 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 References editJohn 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 edit nbsp Wikimedia Commons has media related to Order 6 hexagonal tiling Square tiling Tilings of regular polygons List of uniform planar tilings List of regular polytopesExternal 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 Order 6 hexagonal tiling amp oldid 1189601659, wikipedia, wiki, book, books, library,

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