Triheptagonal tiling

Summary

Triheptagonal tiling
Triheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex configuration (3.7)2
Schläfli symbol r{7,3} or
Wythoff symbol 2 | 7 3
Coxeter diagram CDel node.pngCDel 7.pngCDel node 1.pngCDel 3.pngCDel node.png or CDel node 1.pngCDel split1-73.pngCDel nodes.png
Symmetry group [7,3], (*732)
Dual Order-7-3 rhombille tiling
Properties Vertex-transitive edge-transitive

In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.

Compare to trihexagonal tiling with vertex configuration 3.6.3.6.

ImagesEdit

 
Klein disk model of this tiling preserves straight lines, but distorts angles
 
The dual tiling is called an Order-7-3 rhombille tiling, made from rhombic faces, alternating 3 and 7 per vertex.

7-3 RhombilleEdit

7-3 rhombille tiling
 
FacesRhombi
Coxeter diagram     
Symmetry group[7,3], *732
Rotation group[7,3]+, (732)
Dual polyhedronTriheptagonal tiling
Face configurationV3.7.3.7
Propertiesedge-transitive face-transitive

In geometry, the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane. Sets of three and seven rhombi meet two classes of vertices.

 
7-3 rhombile tiling in band model

Related polyhedra and tilingsEdit

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:

Quasiregular tilings: (3.n)2
Sym.
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco. Noncompact hyperbolic
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
 
*832
[8,3]...
 
*∞32
[∞,3]
 
[12i,3] [9i,3] [6i,3]
Figure
 
                   
Figure
 
       
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.∞)2 (3.12i)2 (3.9i)2 (3.6i)2
Schläfli r{3,3} r{3,4} r{3,5} r{3,6} r{3,7} r{3,8} r{3,∞} r{3,12i} r{3,9i} r{3,6i}
Coxeter
     
    
                                               
                 
Dual uniform figures
Dual
conf.
 
V(3.3)2
 
V(3.4)2
 
V(3.5)2
 
V(3.6)2
 
V(3.7)2
 
V(3.8)2
 
V(3.∞)2

From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.

Uniform heptagonal/triangular tilings
Symmetry: [7,3], (*732) [7,3]+, (732)
                                               
               
{7,3} t{7,3} r{7,3} t{3,7} {3,7} rr{7,3} tr{7,3} sr{7,3}
Uniform duals
                                               
               
V73 V3.14.14 V3.7.3.7 V6.6.7 V37 V3.4.7.4 V4.6.14 V3.3.3.3.7
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 alsoEdit

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.

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