Hexagonal tiling
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| Hexagonal tiling | |
|---|---|
 |
|
| Type | Regular tiling |
| Vertex figure | 6.6.6 |
| Schläfli symbol | {6,3} t{3,6} |
| Wythoff symbol | 3 | 6 2 2 6 | 3 3 3 3 | |
| Coxeter-Dynkin |     |
| Symmetry | p6m |
| Dual | Triangular tiling |
| Properties | Vertex-transitive, edge-transitive, face-transitive |
 6.6.6 |
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In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. It has Schläfli symbol of t0{6,3} or t2{3,6}.
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 square tiling and the triangular tiling.
This hexagonal pattern exists in nature in a beehive's honeycomb.
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[edit] Nonregular forms
The hexagonal tiling can be stretched and adjusted to other geometric proportions and different symmetries. For example, the standard brick pattern can be considered a hexagonal tilings where two pairs of edges have become colinear.
[edit] Uniform colorings
There are 3 distinct uniform colorings of a hexagonal tiling. (Naming the colors by indices on the 3 hexagons around a vertex: 111, 112, 123.)
The 3 colorings, named by their generating Wythoff symbols are:
 3 | 6 2 |
 2 6 | 3 |
 3 3 3 | |
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 |
[edit] 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 (Small rhombitrihexagonal tiling) |
t0,2{6,3} | 6 3 | 2 | 3.4.6.4 |  |
| Omnitruncated hexagonal tiling (Great rhombitrihexagonal 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 |  |
[edit] See also
- Hexagonal lattice
- Hexagonal prismatic honeycomb
- Tilings of regular polygons
- List of uniform tilings
- List of regular polytopes
- Example: Carbon nanotube
- Example: Settlers of Catan
- Example: Chicken wire
[edit] External links
- Eric W. Weisstein, Hexagonal Grid at MathWorld.
[edit] 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
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