Order-6 pentagonal tiling
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| Order-6 pentagonal tiling | |
|---|---|
| Order-6 pentagonal tiling Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 56 |
| Schläfli symbol | {5,6} |
| Wythoff symbol | 6 | 5 2 |
| Coxeter diagram | File:CDel node.pngFile:CDel 6.pngFile:CDel node.pngFile:CDel 5.pngFile:CDel node 1.png |
| Symmetry group | [6,5], (*652) |
| Dual | Order-5 hexagonal tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.
Uniform coloring
This regular tiling can also be constructed from [(5,5,3)] symmetry alternating two colors of pentagons, represented by t1(5,5,3).
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain, and 5 mirrors meeting at a point. This symmetry by orbifold notation is called *33333 with 5 order-3 mirror intersections.
Related polyhedra and tiling
This tiling is topologically related as a part of sequence of regular tilings with order-6 vertices with Schläfli symbol {n,6}, and Coxeter diagram File:CDel node 1.pngFile:CDel n.pngFile:CDel node.pngFile:CDel 6.pngFile:CDel node.png, progressing to infinity.
| [(5,5,3)] reflective symmetry uniform tilings | ||||||
|---|---|---|---|---|---|---|
| File:H2 tiling 355-1.png | File:H2 tiling 355-2.png | File:H2 tiling 355-3.png | File:H2 tiling 355-4.png | File:H2 tiling 355-5.png | File:H2 tiling 355-6.png | File:H2 tiling 355-7.png |
References
- 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.
See also
Wikimedia Commons has media related to Order-6 pentagonal tiling.
