Order-6 hexagonal tiling
Order-6 hexagonal tiling | |
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![]() 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}
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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}
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Spherical | Euclidean | Hyperbolic tilings | ||||||
![]() {6,2} | ![]() {6,3} | ![]() {6,4} | ![]() {6,5} | ![]() {6,6} | ![]() {6,7} | ![]() {6,8} | ... | ![]() {6,∞} |
Uniform hexahexagonal tilings
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Symmetry: [6,6], (*662) | ||||||
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![]() | ![]() | ![]() | ![]() | ![]() | ![]() | ![]() |
{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) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() |
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![]() | ![]() | ![]() | ![]() | ![]() | ||
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
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Coxeter diagrams | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ![]() ![]() ![]() ![]() ![]() | ||||
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![]() ![]() ![]() | ![]() ![]() ![]() | ![]() ![]() ![]() | ![]() ![]() ![]() | |||||
Vertex figure | 66 | (3.4.3.4)2 | 3.4.6.6.4 | 6.4.6.4 | ||||
Image | ![]() | ![]() | ![]() | ![]() | ||||
Dual | ![]() | ![]() |
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
![](http://upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png)
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