Lattice (group)

Summary

A lattice in $\R^n$ is the set of all integer combinations of $n$ linearly independent vectors. Equivalently, it is a discrete subgroup of $\R^n$ that spans $\R^n$. As a group it is free abelian of rank $n$, and its covolume, the volume of a fundamental parallelepiped, measures how dense it is. Lattices are the basic objects of the geometry of numbers, through Minkowski's theorem, and they appear in number fields, in crystallography and in cryptography.