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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 1842Events February 21 John J. Greenough patents the sewing machine. March 5 Over 500 Mexican troops led by Rafael Vasquez invade Texas briefly occupy San Antonio and then head back to the Rio Grande. This is the first such invasion since the Texas Revolution, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1845Events January 29 The Raven by Edgar Allan Poe is published for the first time New York Evening Mirror . March 1 President John Tyler signs a bill authorizing the United States to annex the Republic of Texas. March 3 Florida is admitted as the 27th U..
For a more mathematically intensive discussion, see lattice (group)See lattice for other meanings of this term, both within and without mathematics. In mathematics, a lattice in R n is a discrete subgroup of R n which spans the real vector space R n''. Every lattice in R n can be generated from a basis for the vector spa.
Crystallography