# Lattice of points

point lattice, in $\mathbf R ^ {n}$, with basis $[ \mathbf e _ {1} \dots \mathbf e _ {n} ]$

The set $\Lambda = \mathbf Z {\mathbf e _ {1} } + \dots + \mathbf Z {\mathbf e _ {n} }$ of points $\mathbf a = g _ {1} \mathbf e _ {1} + \dots + g _ {n} \mathbf e _ {n}$, where $g _ {1} \dots g _ {n}$ are integers.

The lattice $\Lambda$ can be regarded as the free Abelian group with $n$ generators. A lattice $\Lambda$ has an infinite number of bases; their general form is $( \mathbf e _ {1} \dots \mathbf e _ {n} ) U$, where $U$ runs through all integral matrices of determinant $\pm 1$. The quantity

$$d ( \Lambda ) = | \mathop{\rm det} ( \mathbf e _ {1} \dots e _ {n} ) | > 0$$

is the volume of the parallelopipedon formed by the vectors $\mathbf e _ {1} \dots \mathbf e _ {n}$. It does not depend on the choice of a basis and is called the determinant of the lattice $\Lambda$.

The partition of point lattices into Voronoi lattice types plays an important role in the geometry of quadratic forms (cf. Quadratic form).