Heptagonal tiling
| 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 | File:CDel node 1.pngFile:CDel 7.pngFile:CDel node.pngFile:CDel 3.pngFile:CDel node.png |
| 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
[edit | edit source]| File:PavageDemiPlanPoincare.svg Poincaré half-plane model |
File:PavageHypPoincare2.svg Poincaré disk model |
File:PavageKleinBeltrami.svg Beltrami-Klein model |
Related polyhedra and tilings
[edit | edit source]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} | |||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Compact hyperb. | Paraco. | Noncompact hyperbolic | |||||||
| File:Spherical trigonal hosohedron.svg | File:Uniform tiling 332-t0-1-.svg | File:Uniform tiling 432-t0.png | File:Uniform tiling 532-t0.png | File:Uniform polyhedron-63-t0.png | File:Heptagonal tiling.svg | File:H2-8-3-dual.svg | File:H2-I-3-dual.svg | File:H2 tiling 23j12-1.png | File:H2 tiling 23j9-1.png | File:H2 tiling 23j6-1.png | File:H2 tiling 23j3-1.png |
| {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.
Hurwitz surfaces
[edit | edit source]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
[edit | edit source]
- Hexagonal tiling
- Tilings of regular polygons
- List of uniform planar tilings
- List of regular polytopes
References
[edit | edit source]- John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value). (Chapter 19, The Hyperbolic Archimedean Tessellations)
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
External links
[edit | edit source]- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- 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
