Торлы нүктелер жиыны және оның геометриялық қасиеттері

Periodic set of points

In geometry and group theory, a lattice in the real coordinate space is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and 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 which spans the vector space For any basis of , 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 are used in various ways in the physical sciences. For instance, in materials science and solid state physics, a lattice is a synonym for the framework 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. More generally, lattice models are studied in physics, often by the techniques of computational physics.

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
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 is the subgroup More complicated examples include the E8 lattice, which is a lattice in , and the Leech lattice in The period lattice in 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.

Minkowski's theorem relates the number d and the volume of a symmetric convex set S to the number of lattice points contained in S. The number of lattice points contained in a polytope all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomial. Formulas for some of the coefficients of this polynomial involve d as well.

Computational lattice problems have many applications in computer science. For example, the Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) has been used in the cryptanalysis of many public key encryption schemes, and many lattice based cryptographic schemes are known to be secure under the assumption that certain lattice problems are computationally difficult.

The 14 lattice types in 3D are called Bravais lattices. They are characterized by their space group. 3D patterns with translational symmetry of a particular type cannot have more, but may have less symmetry than the lattice itself.

A lattice in is a discrete subgroup of which spans as a real vector space. As the dimension of as a real vector space is equal to , a lattice in will be a free abelian group of rank
For example, the Gaussian integers form a lattice in , as is a basis of over .

More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that the quotient G/Γ is of finite measure, for the measure on it inherited from Haar measure on G (left invariant, or right invariant—the definition is independent of that choice). That will certainly be the case when G/Γ is compact, but that sufficient condition is not necessary, as is shown by the case of the modular group in SL2(R), which is a lattice but where the quotient isn't compact (it has cusps). There are general results stating the existence of lattices in Lie groups. A lattice is said to be uniform or cocompact if G/Γ is compact; otherwise the lattice is called non uniform.

A primitive element of a lattice is an element that is not a positive integer multiple of another element in the lattice.