A Bravais lattice is a point lattice, an array generated by integer linear combinations of linearly independent vectors. After choosing a lattice point as the origin, its position vectors in three dimensions have the form
 |
where 
,
, and 
are integers and the vectors 
, 
, and 
form a lattice basis. Every
lattice point has the same environment under translation. Bravais (1850) classified the three-dimensional
cases into 14 types, grouped into seven lattice systems.
The table shows representative unit cells for the 14 types. Blue points mark corners, and orange points
mark additional face or body centers. The drawings are parallel projections
and are independently scaled. The parameters 
, 
,
and 
are the cell-edge lengths, and 
, 
, and 
are the angles between the edges
of lengths 
and 
, 
and 
, and 
and 
, respectively. All lengths are
positive and each unit cell has nonzero volume.
The metric relations are necessary relations for the chosen cells. Special parameter
choices can increase the symmetry and change the Bravais
type (Grimmer 2015, pp. 143-145).
| representative | lattice system | type | cell | metric relations | points per cell |
| triclinic | aP | primitive | none | 1 |
| monoclinic | mP | primitive |  | 1 |
| monoclinic | mS | base-centered |  | 2 |
| orthorhombic | oP | primitive |  | 1 |
| orthorhombic | oS | base-centered |  | 2 |
| orthorhombic | oI | body-centered |  | 2 |
| orthorhombic | oF | face-centered |  | 4 |
| tetragonal | tP | primitive |  ;
 | 1 |
| tetragonal | tI | body-centered |  ;
 | 2 |
| rhombohedral | hR | primitive rhombohedral |  ;  | 1 |
| hexagonal | hP | primitive |  ;
 ,  | 1 |
| cubic | cP | primitive |  ;
 | 1 |
| cubic | cI | body-centered |  ;
 | 2 |
| cubic | cF | face-centered |  ;
 | 4 |
The symbols 
,
, and 
indicate primitive, body-centered, and face-centered cells.
The symbol 
in 
and 
is independent of the choice of centered faces. Here these types are drawn in a 
-centered setting, with added points
at fractional coordinates 
and 
(de Wolff et al. 1985, pp. 278-279).
The last column counts lattice points with boundary sharing included. The eight corners contribute 
in total. A base-centered cell adds 
, a body-centered cell adds 1, and a face-centered cell
adds 
.
The 
row uses a primitive rhombohedral unit cell and therefore
has one effective lattice point. The alternative
hexagonal cell for the same type contains three effective lattice
points. The points represent lattice positions, not
necessarily individual atoms.
