Hexagonal lattice
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Hexagonal lattice | Wallpaper group p6m | Unit cell |
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The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types.[1] The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths,
- [math]\displaystyle{ |\mathbf a_1| = |\mathbf a_2| = a. }[/math]
The reciprocal lattice of the hexagonal lattice is a hexagonal lattice in reciprocal space with orientation changed by 90° and primitive lattice vectors of length
- [math]\displaystyle{ g=\frac{4\pi}{a\sqrt 3}. }[/math]
Honeycomb point set
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis.[1] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices.
In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb point set.
Crystal classes
The hexagonal lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
Geometric class, point group | Arithmetic class[further explanation needed] |
Wallpaper groups | ||||
---|---|---|---|---|---|---|
Schön. | Intl | Orb. | Cox. | |||
C3 | 3 | (33) | [3]+ | None | p3 (333) |
|
D3 | 3m | (*33) | [3] | Between | p3m1 (*333) |
p31m (3*3) |
C6 | 6 | (66) | [6]+ | None | p6 (632) |
|
D6 | 6mm | (*66) | [6] | Both | p6m (*632) |
See also
- Square lattice
- Hexagonal tiling
- Close-packing
- Centered hexagonal number
- Eisenstein integer
- Voronoi diagram
- Hermite constant
References
![]() | Original source: https://en.wikipedia.org/wiki/Hexagonal lattice.
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