# Hexagonal lattice

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,

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

$g={\frac {4\pi }{a{\sqrt {3}}}}.$ ## Honeycomb point set Honeycomb point set as a hexagonal lattice with a two-atom basis. The gray rhombus is a primitive cell. Vectors $\mathbf {a} _{1}$ and $\mathbf {a} _{2}$ are primitive translation vectors.

The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. 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. Intl Orb. Cox.
C3 3 (33) + None p3
(333)

D3 3m (*33)  Between p3m1
(*333)
p31m
(3*3)
C6 6 (66) + None p6
(632)

D6 6mm (*66)  Both p6m
(*632)