In geometry and group theory, a lattice in the real coordinate space R n {\displaystyle \mathbb {R} ^{n}} is an infinite set of lattice points having the following properties:
Coordinate-wise addition or subtraction of two points in the lattice produces another lattice point. The lattice points are all separated by some minimum distance (isolated). Every point in the space is within some maximum distance of a lattice point. One of the simplest examples of a lattice is the square lattice, which consists of all points ( a , b ) {\displaystyle (a,b)} in the plane whose coordinates are both integers, and its higher-dimensional analogues, the integer lattices Z n {\displaystyle \mathbb {Z} ^{n}} . Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space. The requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n {\displaystyle n} which spans the vector space R n {\displaystyle \mathbb {R} ^{n}} . For any basis of R n {\displaystyle \mathbb {R} ^{n}} , the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from a basis in this way. 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 percolation theory to study connectivity arising from small-scale interactions, cryptography because of conjectured computational hardness of several lattice problems, and occur frequently in the physical sciences. For instance, in materials science and solid-state physics, a lattice is a synonym for a crystalline structure, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or molecule positions in a crystal. More generally, lattice models are studied in physics, often by the techniques of computational physics.
Symmetry considerations and examples A lattice is the symmetry group of discrete translational symmetry in n directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. As a group (dropping its geometric structure) a lattice is a finitely generated free abelian group, and thus isomorphic to Z n {\displaystyle \mathbb {Z} ^{n}} . A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense. A simple example of a lattice in R n {\displaystyle \mathbb {R} ^{n}} is the subgroup Z n {\displaystyle \mathbb {Z} ^{n}} . More complicated examples include the E8 lattice, which is a lattice in R 8 {\displaystyle \mathbb {R} ^{8}} , and the Leech lattice in R 24 {\displaystyle \mathbb {R} ^{24}} . The period lattice in R 2 {\displaystyle \mathbb {R} ^{2}} is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalizes to higher dimensions in the theory of abelian functions. Lattices called root lattices are important in the theory of simple Lie algebras; for example, the E8 lattice is related to a Lie algebra that goes by the same name.
Lattice basis tiling space A lattice Λ {\displaystyle \Lambda } in R n {\displaystyle \mathbb {R} ^{n}} thus has the form
Λ = { ∑ i = 1 n a i v i | a i ∈ Z } , {\displaystyle \Lambda ={\biggl \{}\sum _{i=1}^{n}a_{i}v_{i}\mathbin {\bigg \vert } a_{i}\in \mathbb {Z} {\biggr \}},}
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