Hexagonal lattice

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,

Not to be confused with Hexagonal crystal family.
Hexagonal lattice Wallpaper group p6m Unit cell

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

Honeycomb point set
Honeycomb point set as a hexagonal lattice with a two-atom basis. The gray rhombus is a primitive cell. Vectors and are primitive translation vectors.

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		 Wallpaper groups
Schön.IntlOrb.Cox.
C33(33)[3]+ None p3
(333)		  
D33m(*33)[3] Between p3m1
(*333)		 p31m
(3*3)
C66(66)[6]+ None p6
(632)		  
D66mm(*66)[6] Both p6m
(*632)		  

See also

References
  1. Rana, Farhan. "Lattices in 1D, 2D, and 3D" (PDF). Cornell University. Archived (PDF) from the original on 2020-12-18.
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