Lattice (group)

:See lattice for other meanings of this term, both within and without mathematics.

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In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integral coefficients.

Related Topics:
Mathematics - Geometry - Group theory - Discrete - Subgroup - Spans - Real - Vector space - Basis - Linear combination - Integral

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A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is very similar, but that is in general a translate which need not contain the origin.

Related Topics:
Dimension - Atom - Molecule - Crystal - Group action

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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.

Related Topics:
Symmetry group - Translational symmetry

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A simple example of a lattice in Rn is the subgroup Zn. A more complicated example is the Leech lattice, which is a lattice in R24. The period lattice in R2 is central to the study of elliptic functions, developed in nineteenth century mathematics; it generalises to higher dimensions in the theory of abelian functions.

Related Topics:
Leech lattice - Period lattice - Elliptic functions - Nineteenth century - Abelian function

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A typical lattice Λ in Rn thus has the form

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:

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Lambda = left{ sum_{i=1}^n a_i v_i ; | ; a_i inBbb{Z} ight}

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where {v1, ..., vn} is a basis for Rn. Different bases can generate the same lattice, but the absolute value of the determinant of the vectors vi is uniquely determined by Λ, and is denoted by d(Λ).

Related Topics:
Absolute value - Determinant

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If one thinks of a lattice as dividing the whole of Rn into equal polyhedra (copies of an n-dimensional parallelepiped, known as the fundamental region of the lattice), then d(Λ) is equal to the n-dimensional volume of this polyhedron. This is why d(Λ) is sometimes called the covolume of the lattice.

Related Topics:
Polyhedra - Parallelepiped - Fundamental region - Volume

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Minkowski's theorem relates the number d(Λ) 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(Λ) as well.

Related Topics:
Minkowski's theorem - Convex - Polytope - Ehrhart polynomial

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Lattice basis reduction is the problem of finding a short lattice basis. The Lenstra-Lenstra-Lovász lattice reduction algorithm (LLL) finds a short lattice basis in polynomial time; it has found numerous applications, particularly in public-key cryptography.

Related Topics:
Lenstra-Lenstra-Lovász lattice reduction algorithm - Polynomial time - Public-key cryptography

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A lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn.

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For example, the Gaussian integers form a lattice in C.

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Every lattice in Rn is a free abelian group of rank n; every lattice in Cn is a free abelian group of rank 2n.

Related Topics:
Free abelian group - Rank

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This concept is used in materials science, in which a lattice is a 3-dimensional array of regularly spaced points coinciding with the atom or molecule positions in a crystal.

Related Topics:
Materials science - Atom - Molecule - Crystal

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It also occurs in computational physics, in which a lattice is an n-dimensional geometrical structure of sites, connected by bonds, which represent positions which may be occupied by atoms, molecules, electrons, spins, etc. For an article dealing with the formal representation of such structures see Lattice Geometries. Quite general lattice models are used in physics.

Related Topics:
Lattice model - Physics

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