Hexagonal lattice

Not to be confused with Hexagonal crystal family.

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,

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

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

${\displaystyle g={\frac {4\pi }{a{\sqrt {3}}}}.}$

Honeycomb point set

Honeycomb point set as a hexagonal lattice with a two-atom basis. The gray rhombus is a primitive cell. Vectors ${\displaystyle \mathbf {a} _{1}}$ and ${\displaystyle \mathbf {a} _{2}}$ 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				Wallpaper groups
Schön.	Intl	Orb.	Cox.
C3	3	(33)	[3]+	p3 (333)
D3	3m	(*33)	[3]	p3m1 (*333)	p31m (3*3)
C6	6	(66)	[6]+	p6 (632)
D6	6mm	(*66)	[6]	p6m (*632)

See also

• Square lattice
• Hexagonal tiling
• Close-packing
• Centered hexagonal number
• Eisenstein integer
• Voronoi diagram
• Hermite constant

References

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