Heptagonal tiling
| Heptagonal tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 73 |
| Schläfli symbol | {7,3} |
| Wythoff symbol | 3 | 7 2 |
| Coxeter diagram |    
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| Symmetry group | [7,3], (*732) |
| Dual | Order-7 triangular tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, a 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
[edit] Poincaré half-plane model |
 Poincaré disk model |
 Beltrami-Klein model |
Related polyhedra and tilings
[edit]This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.
| *n32 symmetry mutation of regular tilings: {n,3} | |||||||||||
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| Spherical | Euclidean | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
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| {2,3} | {3,3} | {4,3} | {5,3} | {6,3} | {7,3} | {8,3} | {∞,3} | {12i,3} | {9i,3} | {6i,3} | {3i,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 | |||||||||||
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| Symmetry: [7,3], (*732) | [7,3]+, (732) | ||||||||||
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| {7,3} | t{7,3} | r{7,3} | t{3,7} | {3,7} | rr{7,3} | tr{7,3} | sr{7,3} | ||||
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| 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
[edit]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
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- Hexagonal tiling
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
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 Archived 2013-03-24 at the Wayback Machine
- KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
- Hyperbolic Planar Tessellations, Don Hatch
