Hexagonal lattice

Not to be confused with Hexagonal crystal family.

Two-dimensional Bravais lattices:
1 – oblique (monoclinic),
2 – rectangular (orthorhombic),
3 – centered rectangular (orthorhombic),
4 – hexagonal,
5 – square (tetragonal).

The hexagonal lattice or 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 lattice

Honeycomb lattice 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 lattice 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 lattice can be seen as the union of two offset triangular lattices.

In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb lattice.

Crystal classes

The hexagonal lattice class names, Schönflies notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.

Geometric class, point group				Arithmetic class	Wallpaper groups
Schön.	Orbifold	Cox.	Ord.
C3	(33)	[3]+	3	None	p3 (333)
D3	(*33)	[3]	6	Between	p3m1 (*333)	p31m (3*3)
C6	(66)	[6]+	6	None	p6 (632)
D6	(*66)	[6]	12	Both	p6m (*632)

See also

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