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Triheptagonal tiling

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Title: Triheptagonal tiling  
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Triheptagonal tiling

Triheptagonal tiling
Triheptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic uniform tiling
Vertex figure 3.7.3.7
Schläfli symbol r{7,3}
Wythoff symbol 2 | 7 3
Coxeter diagram
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.

Images


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.

Related polyhedra and tilings

The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:
Dimensional family of quasiregular polyhedra and tilings: 3.n.3.n
Symmetry
*n32
[n,3]
Spherical Euclidean Compact hyperbolic Paracompact Noncompact
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*632
[6,3]
p6m
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
 
[iπ/λ,3]
Quasiregular
figures
configuration

3.3.3.3

3.4.3.4

3.5.3.5

3.6.3.6

3.7.3.7

3.8.3.8

3.∞.3.∞
3.∞.3.∞
Coxeter diagram
Dual
(rhombic)
figures
configuration

V3.3.3.3

V3.4.3.4

V3.5.3.5

V3.6.3.6

V3.7.3.7

V3.8.3.8

V3.∞.3.∞
Coxeter diagram

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} 2t{7,3}=t{3,7} 2r{7,3}={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 also

References

  • 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.  

External links

  • 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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