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In mathematics, 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 considering all linear combinations with integral coefficients.
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.
A typical lattice L in Rn thus has the form
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 L, and is denoted by d(L). If one thinks of a lattice as dividing the whole of Rn into equal polyhedra (known as the fundamental region of the lattice), then d(L) is equal to the n-dimensional volumeVolume (also called capacity is a quantification of how much space an object occupies. The SI unit for volume is the cubic metre (American spelling meter). The volume of a solid object is a numerical value given to describe the three-dimensional concept o of this polyhedron. This is why d(L) is sometimes called the covolume of the lattice.
Minkowski's theoremGeometry of numbers Theorems In mathematics, Minkowski's theorem in the geometry of numbers applies to convex symmetric sets and lattices; it relates the number of contained lattice points to the volume of such a set. This relationship was discovered by H relates the number d(L) and the volume of a symmetric convexIn mathematics, an object is convex if for any pair of points within the object, any point on the straight line segment that joins them is also within the object. For example, a solid cube is convex, but anything that is hollow or has a dent in it is not set S to the number of lattice points contained in S. The number of lattice points contained in a polytopeIn geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. Beyond that, the term is used for a variety of related mathematical concepts. This is analogous to the way the term sq all of whose vertices are elements of the lattice is described by the polytope's Ehrhart polynomialIn mathematics integral polytopes have associated Ehrhart polynomials which encode some geometrical information about them. Specifically, consider a lattice L in Euclidean space R n and an n dimensional polytope P in R n and assume that all vertices of th. Formulas for some of the coefficients of this polynomial involve d(L) as well.A lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn. For example, the Gaussian integerA Gaussian integer is a complex number whose real and imaginary part are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as Z ''i . This is a Euclidean domain whis form a lattice in C.
Every lattice in Rn is a free abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Unlike of rankIn mathematics, the rank or torsion-free rank of an abelian group measures how large a group is in terms of how large a vector space one would need to "contain" it; or alternatively how large a free abelian group it can contain as a subgroup. Definition A n; every lattice in Cn is a free abelian group of rank 2n.
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.
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.
More generally, a lattice Γ in a Lie group G is a discrete subgroup, such that 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.
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