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Hexagonal lattice
One of the five 2D Bravais lattices
Hexagonal latticeWallpaper group p6mUnit cell

The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional Bravais lattice types. 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,

| a 1 | = | a 2 | = a . {\displaystyle |\mathbf {a} _{1}|=|\mathbf {a} _{2}|=a.}

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

g = 4 π a 3 . {\displaystyle g={\frac {4\pi }{a{\sqrt {3}}}}.}
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Honeycomb point set

The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis.2 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 groupWallpaper groups
Schön.IntlOrb.Cox.
C33(33)[3]+p3(333) 
D33m(*33)[3]p3m1(*333)p31m(3*3)
C66(66)[6]+p6(632) 
D66mm(*66)[6]p6m(*632) 

See also

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References

  1. Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18. https://courses.cit.cornell.edu/ece407/Lectures/handout4.pdf

  2. Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18. https://courses.cit.cornell.edu/ece407/Lectures/handout4.pdf