In mathematics, a lattice is a periodic arrangement of points such as the vertices of a tiling of space by cubes or the positions of atoms in a crystal. More technically, a discrete Abelian subgroup of an n-dimensional vector space which not contained in an (n – 1)-dimensional vector space. Lattices play a central role in the theory of Lie groups, in number theory, in error-correcting codes, and many other areas of mathematics. A lattice is often, but not always, distinguished from a graph in that a lattice is a graph with a regular structure.

A lattice path is a sequence of points in a lattice such that each point differs from its predecessor by a finite list of allowed steps. Random lattice paths are an interesting model for the random motion of a particle and lattice paths are also important in enumerative combinatorics.

 

A lattice point is a point with integer coordinates.

 

In chemistry, a lattice is a regular network of fixed points about which molecules, atoms, or ions vibrate in a crystal.