Bravais lattice

In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations. A Bravais lattice looks exactly the same no matter from which point one views it.

The position vectors of a Bravais lattice in three dimensions are given by

where n₁, n₂, and n₃ are integers and a₁, a₂, and a₃ are three non-coplanar vectors, called primitive vectors.

When classified by space group, there are 14 unique Bravais lattices in three dimensions. These can be grouped according to their crystal system, or crystallographic point group. The 14 Bravais lattices are:

Crystal system	lattice
triclinic
monoclinic	simple
	centered
orthorhombic	simple
	base-centered
	body-centered
	face-centered
hexagonal
rhombohedral (trigonal)
tetragonal	simple
	body-centered
cubic (isometric)	simple
	body-centered
	face-centered

The Bravais lattices were studied by M. L. Frankenheim in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

See also

For a more mathematically intensive discussion, see lattice (group).