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Tetraheptagonal tiling

December 12, 2023

Tetraheptagonal tiling

Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (4.7)2
Schläfli symbol r{7,4} or
rr{7,7}
Wythoff symbol 2 | 7 4
7 7 | 2
Coxeter diagram
Symmetry group [7,4], (*742)
[7,7], (*772)
Dual Order-7-4 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the tetraheptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of r{4,7}.

Symmetry edit

 
A half symmetry [1+,4,7] = [7,7] construction exists, which can be seen as two colors of heptagons. This coloring can be called a rhombiheptaheptagonal tiling. 		 
The dual tiling is made of rhombic faces and has a face configuration V4.7.4.7.

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
Uniform heptagonal/square tilings
Symmetry: [7,4], (*742) [7,4]+, (742) [7+,4], (7*2) [7,4,1+], (*772)
                                                           
                   
{7,4} t{7,4} r{7,4} 2t{7,4}=t{4,7} 2r{7,4}={4,7} rr{7,4} tr{7,4} sr{7,4} s{7,4} h{4,7}
Uniform duals
                                                           
               
V74 V4.14.14 V4.7.4.7 V7.8.8 V47 V4.4.7.4 V4.8.14 V3.3.4.3.7 V3.3.7.3.7 V77
Uniform heptaheptagonal tilings
Symmetry: [7,7], (*772) [7,7]+, (772)
      =    
=       		      =    
=       		      =    
=       		      =    
=       		      =    
=       		      =    
=       		      =   
=       		      =   
=      
               
{7,7} t{7,7}
 r{7,7} 2t{7,7}=t{7,7} 2r{7,7}={7,7} rr{7,7} tr{7,7} sr{7,7}
Uniform duals
                                               
             
V77 V7.14.14 V7.7.7.7 V7.14.14 V77 V4.7.4.7 V4.14.14 V3.3.7.3.7
Dimensional family of quasiregular polyhedra and tilings: 7.n.7.n
Symmetry
*7n2
[n,7] 		Hyperbolic... Paracompact Noncompact
*732
[3,7] 		*742
[4,7] 		*752
[5,7] 		*762
[6,7] 		*772
[7,7] 		*872
[8,7]... 		*∞72
[∞,7] 		 
[iπ/λ,7]
Coxeter                                                
Quasiregular
figures
configuration 		 
3.7.3.7 		 
4.7.4.7 		 
7.5.7.5 		 
7.6.7.6 		 
7.7.7.7 		 
7.8.7.8 		 
7.∞.7.∞ 		 
7.∞.7.∞

See also edit

 
Wikimedia Commons has media related to Uniform tiling 4-7-4-7.

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.

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


 

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December 12, 2023
tetraheptagonal, tiling, poincaré, disk, model, hyperbolic, planetype, hyperbolic, uniform, tilingvertex, configuration, 2schläfli, symbol, displaystyle, begin, bmatrix, bmatrix, wythoff, symbol, 2coxeter, diagramsymmetry, group, dual, order, rhombille, tiling. Tetraheptagonal tilingPoincare disk model of the hyperbolic planeType Hyperbolic uniform tilingVertex configuration 4 7 2Schlafli symbol r 7 4 or 7 4 displaystyle begin Bmatrix 7 4 end Bmatrix rr 7 7 Wythoff symbol 2 7 47 7 2Coxeter diagramSymmetry group 7 4 742 7 7 772 Dual Order 7 4 rhombille tilingProperties Vertex transitive edge transitiveIn geometry the tetraheptagonal tiling is a uniform tiling of the hyperbolic plane It has Schlafli symbol of r 4 7 Contents 1 Symmetry 2 Related polyhedra and tiling 3 See also 4 References 5 External linksSymmetry edit nbsp A half symmetry 1 4 7 7 7 construction exists which can be seen as two colors of heptagons This coloring can be called a rhombiheptaheptagonal tiling nbsp The dual tiling is made of rhombic faces and has a face configuration V4 7 4 7 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 2Uniform heptagonal square tilings vteSymmetry 7 4 742 7 4 742 7 4 7 2 7 4 1 772 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 4 t 7 4 r 7 4 2t 7 4 t 4 7 2r 7 4 4 7 rr 7 4 tr 7 4 sr 7 4 s 7 4 h 4 7 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 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp V74 V4 14 14 V4 7 4 7 V7 8 8 V47 V4 4 7 4 V4 8 14 V3 3 4 3 7 V3 3 7 3 7 V77Uniform heptaheptagonal tilings vteSymmetry 7 7 772 7 7 772 nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp 7 7 t 7 7 r 7 7 2t 7 7 t 7 7 2r 7 7 7 7 rr 7 7 tr 7 7 sr 7 7 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 nbsp nbsp nbsp nbsp nbsp V77 V7 14 14 V7 7 7 7 V7 14 14 V77 V4 7 4 7 V4 14 14 V3 3 7 3 7Dimensional family of quasiregular polyhedra and tilings 7 n 7 n vteSymmetry 7n2 n 7 Hyperbolic Paracompact Noncompact 732 3 7 742 4 7 752 5 7 762 6 7 772 7 7 872 8 7 72 7 ip l 7 Coxeter nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp nbsp Quasiregularfiguresconfiguration nbsp 3 7 3 7 nbsp 4 7 4 7 nbsp 7 5 7 5 nbsp 7 6 7 6 nbsp 7 7 7 7 nbsp 7 8 7 8 nbsp 7 7 7 7 See also edit nbsp Wikimedia Commons has media related to Uniform tiling 4 7 4 7 Uniform tilings in hyperbolic plane List of regular polytopesReferences 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 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 nbsp This hyperbolic geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Tetraheptagonal tiling amp oldid 1189586162, wikipedia, wiki, book, books, library,

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