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 |
Klein quartic
A quotient of the order-7 triangular tiling yields the Klein quartic, a highly symmetric genus 3 surface, together with a tiling by 20 triangles, meeting at 24 vertices. The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron.[1]
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)
- 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
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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