Heptagonal tiling
Tiling of the hyperbolic plane
| 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 | |
| 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.
01Images
03Hurwitz surfaces
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.
Sources and credits
This article is adapted from the Wikipedia article “Heptagonal tiling”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.
Images, from Wikimedia Commons:
- Heptagonal tiling.svg by Parcly Taxel, Public domain
- 3-7 kisrhombille.svg by Parcly Taxel, Public domain
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