Order-6 pentagonal tiling
In geometry, the order-6 pentagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {5,6}.
Order-6 pentagonal tiling | |
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![]() 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 | ![]() ![]() ![]() ![]() ![]() |
Symmetry group | [6,5], (*652) |
Dual | Order-5 hexagonal tiling |
Properties | Vertex-transitive, edge-transitive, face-transitive |
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 , progressing to infinity.
Regular tilings {n,6} | ||||||||
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Spherical | Euclidean | Hyperbolic tilings | ||||||
![]() {2,6} ![]() ![]() ![]() ![]() ![]() |
![]() {3,6} ![]() ![]() ![]() ![]() ![]() |
![]() {4,6} ![]() ![]() ![]() ![]() ![]() |
![]() {5,6} ![]() ![]() ![]() ![]() ![]() |
![]() {6,6} ![]() ![]() ![]() ![]() ![]() |
![]() {7,6} ![]() ![]() ![]() ![]() ![]() |
![]() {8,6} ![]() ![]() ![]() ![]() ![]() |
... | ![]() {∞,6} ![]() ![]() ![]() ![]() ![]() |
Uniform hexagonal/pentagonal tilings | |||||||||||
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Symmetry: [6,5], (*652) | [6,5]+, (652) | [6,5+], (5*3) | [1+,6,5], (*553) | ||||||||
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{6,5} | t{6,5} | r{6,5} | 2t{6,5}=t{5,6} | 2r{6,5}={5,6} | rr{6,5} | tr{6,5} | sr{6,5} | s{5,6} | h{6,5} | ||
Uniform duals | |||||||||||
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V65 | V5.12.12 | V5.6.5.6 | V6.10.10 | V56 | V4.5.4.6 | V4.10.12 | V3.3.5.3.6 | V3.3.3.5.3.5 | V(3.5)5 |
[(5,5,3)] reflective symmetry uniform tilings | ||||||
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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)
- "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
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