A lattice in $\mathbb R^n$ is a discrete subgroup of $\mathbb R^n$ or, equivalently, it is a subgroup of $\mathbb R^n$ generated by linearly independent vectors. Lattices have applications in geometric number theory, e.g. via Minkowski's theorem.
In geometry and group theory, a lattice in $\Bbb R^n$ is a subgroup of the additive group $\Bbb R^n$ that is isomorphic to the additive group $\Bbb Z^n$ and that spans the real vector space $\Bbb R^n$. In other words, for any basis of $\Bbb R^n$, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice. A lattice may be viewed as a regular tiling of a space by a primitive cell.
Lattices have many significant applications in pure mathematics, particularly in connection to Lie algebras, number theory, and group theory. They also arise in applied mathematics in connection with coding theory, in cryptography because of conjectured computational hardness of several lattice problems, and in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for the "frame work" of a crystalline structure, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or molecule positions in a crystal.
