| Hexagonal tiling | |
|---|---|
 |
|
| Type | Regular tiling |
| Vertex figure | 6.6.6 (or 63) |
| Schläfli symbol(s) | {6,3} t0,1{3,6} |
| Wythoff symbol(s) | 3 | 6 2 2 6 | 3 3 3 3 | |
| Coxeter-Dynkin(s) |      |
| Symmetry | *632 |
| Dual | Triangular tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
 6.6.6 (or 63) |
|
In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol of {6,3} or t{3,6} (as a truncated triangular tiling).
Conway calls it a hextille.
The internal angle of the hexagon is 120 degrees so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling.
This hexagonal pattern exists in nature in a beehive's honeycomb, and various crystal lattices.
Contents |
Uniform colorings
There are 3 distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions.
| Coloring |  |
 |
 |
|---|---|---|---|
| Schläfli symbol | {6,3} | t{3,6} | |
| Wythoff symbol | 3 | 6 2 | 2 6 | 3 | 3 3 3 | |
| symmetry | *632 (p6m) | *632 (p6m) | *333 (p3) |
| Coxeter-Dynkin diagram | |
|
|
The 3-color tiling is a tessellation generated by the order-3 permutohedrons.
Topologically identical tilings
The hexagonal tiling can be stretched and adjusted to other geometric proportions and different symmetries.
The standard brick pattern can be considered a nonregular hexagonal tiling. Each rectangular brick has vertices inserted on the two long edges, dividing them into two colinear edges.
It can also be distorted into a chiral 4-colored tri-directional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 4-colored faces have rotational 632 (p6) symmetry.
The Herringbone pattern is also a distorted hexagonal tiling.
 Hexagonal weave |
 Herringbone |
Related polyhedra and tilings
This tiling is topologically related as a part of sequence of regular polyhedra with vertex figure (n3), and continue into the hyperbolic plane.
 (33) |
 (43) |
 (53) |
 (63) tiling |
 (73) tiling |
It is also topologically related as a part of sequence of uniform truncated polyhedra with vertex figure (n.6.6).
 (3.6.6) |
 (4.6.6) |
 (5.6.6) |
 (6.6.6) tiling |
 (7.6.6) tiling |
Wythoff constructions from hexagonal and triangular tilings
Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)
| Tiling | Schläfli symbol |
Wythoff symbol |
Vertex figure |
Image |
|---|---|---|---|---|
| Hexagonal tiling | t0{6,3} | 3 | 6 2 | 63 | |
| Truncated hexagonal tiling | t0,1{6,3} | 2 3 | 6 | 3.12.12 |  |
| Rectified hexagonal tiling (Trihexagonal tiling) |
t1{6,3} | 2 | 6 3 | (3.6)2 |  |
| Bitruncated hexagonal tiling (Truncated triangular tiling) |
t1,2{6,3} | 2 6 | 3 | 6.6.6 | |
| Dual hexagonal tiling (Triangular tiling) |
t2{6,3} | 6 | 3 2 | 36 |  |
| Cantellated hexagonal tiling (Rhombitrihexagonal tiling) |
t0,2{6,3} | 6 3 | 2 | 3.4.6.4 |  |
| Omnitruncated hexagonal tiling (Truncated trihexagonal tiling) |
t0,1,2{6,3} | 6 3 2 | | 4.6.12 |  |
| Snub hexagonal tiling | s{6,3} | | 6 3 2 | 3.3.3.3.6 |  |
See also
- Hexagonal lattice
- Hexagonal prismatic honeycomb
- Tilings of regular polygons
- List of uniform tilings
- List of regular polytopes
- Example: Carbon nanotube
- Example: Chicken wire
References
- Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p.296, Table II: Regular honeycombs
- Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-716-71193-1. (Chapter 2.1: Regular and uniform tilings, p.58-65)
- Williams, Robert The Geometrical Foundation of Natural Structure: A Source Book of Design New York: Dover, 1979. p35
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]
External links
- Weisstein, Eric W., "Hexagonal Grid" from MathWorld.
- Weisstein, Eric W., "Regular tessellation" from MathWorld.
- Weisstein, Eric W., "Uniform tessellation" from MathWorld.
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