# lattice

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.