Hexagonal lattice

Short description: One of the five 2D Bravais lattices

Hexagonal lattice	Wallpaper group p6m	Unit 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,

[math]\displaystyle{ |\mathbf a_1| = |\mathbf a_2| = a. }[/math]

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

[math]\displaystyle{ g=\frac{4\pi}{a\sqrt 3}. }[/math]

Honeycomb point set

Honeycomb point set as a hexagonal lattice with a two-atom basis. The gray rhombus is a primitive cell. Vectors [math]\displaystyle{ \mathbf a_1 }[/math] and [math]\displaystyle{ \mathbf a_2 }[/math] 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)	[3]+	None	p3 (333)
D3	3m	(*33)	[3]	Between	p3m1 (*333)	p31m (3*3)
C6	6	(66)	[6]+	None	p6 (632)
D6	6mm	(*66)	[6]	Both	p6m (*632)

See also

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