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.