Order-7 triangular tiling
| Order-7 triangular tiling | |
|---|---|
 Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 37 |
| Schläfli symbol | {3,7} |
| Wythoff symbol | 7 | 3 2 |
| Coxeter diagram |    
|
| Symmetry group | [7,3], (*732) |
| Dual | Heptagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}.
 {3,3} |
 {3,4} |
 {3,5} |
|
 {3,6} |
 {3,7} |
 {3,8} |
 {3,9} |
The dual tiling is the order-3 heptagonal tiling.
 order-3 heptagonal tiling |
order-7 triangular tiling |
Wythoff constructions yields further uniform tilings, yielding eight uniform tilings, including the two regular ones given here.
Hurwitz 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 triangulation whose symmetry group equals their automorphism group as Riemann surfaces.
The smallest of these is the Klein quartic, the most symmetric genus 3 surface, together with a tiling by 56 triangles, meeting at 24 vertices, with symmetry group the simple group of order 168, known as PSL(2,7). The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron.[1]
The dual order-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces.
References
- ^ a b (Richter) Note each face in the polyhedron consist of multiple faces in the tiling – two triangular faces constitute a square face and so forth, as per this explanatory image.
- 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)
- The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-0 – 0, ISBN 0-486-40919-8 (Chapter 10: Regular honeycombs in hyperbolic space)
- Richter, David A., How to Make the Mathieu Group M24, retrieved 2010-04-15
{{citation}}: CS1 maint: ref duplicates default (link)
See also
- List of regular polytopes
- List of uniform planar tilings
- Tilings of regular polygons
- Triangular tiling
- Uniform tilings in hyperbolic plane
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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