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Heptagonal tiling

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Subject: Uniform tilings in hyperbolic plane, Order-7 triangular tiling, Truncated triheptagonal tiling, Truncated order-7 triangular tiling, Order-4 heptagonal tiling
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Heptagonal tiling

Heptagonal tiling
Heptagonal tiling
Poincaré disk model of the hyperbolic plane
Type Hyperbolic regular tiling
Vertex figure 7.7.7
Schläfli symbol {7,3}
Wythoff symbol 3 | 7 2
Coxeter diagram
Symmetry group [7,3], (*732)
Dual Order-7 triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive

In geometry, the heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons around each vertex.

Images


Poincaré half-plane model

Poincaré disk model

Klein-Beltrami model

Related polyhedra and tilings

This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli_symbol {n,3}.
Spherical
Polyhedra
Polyhedra Euclidean Hyperbolic tilings

{2,3}

{3,3}

{4,3}

{5,3}

{6,3}

{7,3}

{8,3}
...
(∞,3}

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

Hurwitz surfaces

The symmetry group of the heptagonal tiling has fundamental domain the (2,3,7) Schwarz triangle, which yields this tiling.

The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a tiling by heptagons whose symmetry group equals their automorphism group as Riemann surfaces. The smallest Hurwitz surface is the Klein quartic (genus 3, automorphism group of order 168), and the induced tiling has 24 heptagons, meeting at 56 vertices.

The dual order-7 triangular tiling has the same symmetry group, and thus yields triangulations of Hurwitz surfaces.

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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