| Order-6 hexagonal tiling | |
|---|---|
| Poincaré disk model of the hyperbolic plane | |
| Type | Hyperbolic regular tiling |
| Vertex configuration | 66 |
| Schläfli symbol | {6,6} |
| Wythoff symbol | 6 | 6 2 |
| Coxeter diagram | |
| Symmetry group | [6,6], (*662) |
| Dual | self dual |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
In geometry, the order-6 hexagonal tiling is a regular tiling of the hyperbolic plane. It has Schläfli symbol of {6,6} and is self-dual.
Symmetry
This tiling represents a hyperbolic kaleidoscope of 6 mirrors defining a regular hexagon fundamental domain. This symmetry by orbifold notation is called *333333 with 6 order-3 mirror intersections. In Coxeter notation can be represented as [6*,6], removing two of three mirrors (passing through the hexagon center) in the [6,6] symmetry.
The even/odd fundamental domains of this kaleidoscope can be seen in the alternating colorings of the tiling:
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}
| ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | ||||||
| {2,6} | {3,6} | {4,6} | {5,6} | {6,6} | {7,6} | {8,6} | ... | {∞,6} |
This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.
*n62 symmetry mutation of regular tilings: {6,n}
| ||||||||
|---|---|---|---|---|---|---|---|---|
| Spherical | Euclidean | Hyperbolic tilings | ||||||
| {6,2} | {6,3} | {6,4} | {6,5} | {6,6} | {6,7} | {6,8} | ... | {6,∞} |
Uniform hexahexagonal tilings
| ||||||
|---|---|---|---|---|---|---|
| Symmetry: [6,6], (*662) | ||||||
| = = | = = | = = | = = | = = | = = | == |
| {6,6}= h{4,6} | t{6,6}= h2{4,6} | r{6,6}{6,4} | t{6,6}= h2{4,6} | {6,6}= h{4,6} | rr{6,6}r{6,4} | tr{6,6}t{6,4} |
| Uniform duals | ||||||
| V66 | V6.12.12 | V6.6.6.6 | V6.12.12 | V66 | V4.6.4.6 | V4.12.12 |
| Alternations | ||||||
| [1+,6,6](*663) | [6+,6](6*3) | [6,1+,6](*3232) | [6,6+](6*3) | [6,6,1+](*663) | [(6,6,2+)](2*33) | [6,6]+(662) |
| = | = | = | ||||
| h{6,6} | s{6,6} | hr{6,6} | s{6,6} | h{6,6} | hrr{6,6} | sr{6,6} |
Similar H2 tilings in *3232 symmetry
| ||||||||
|---|---|---|---|---|---|---|---|---|
| Coxeterdiagrams | ||||||||
| Vertexfigure | 66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
| Image | ||||||||
| Dual | ||||||||
- 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.