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Ingeometry andgroup theory, alattice in thereal coordinate space${\displaystyle {\mathbb{R}}^n}$ is an infinite set of points in this space with these properties:^[1]

• 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.
• Every point in the space is within some maximum distance of a lattice point.

One of the simplest examples of a lattice is thesquare lattice, which consists of all points${\displaystyle (a,b)}$ in the plane whose coordinates are bothintegers, and its higher-dimensional analogues theinteger lattices${\displaystyle {\mathbb{Z}}^n}$.

Closure under addition and subtraction means that a lattice must be asubgroup 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 aDelone set.^[2]

More abstractly, a lattice can be described as afree abelian group of dimension${\displaystyle n}$ whichspans thevector space⁠${\displaystyle {\mathbb{R}}^n}$⁠. For anybasis of⁠${\displaystyle {\mathbb{R}}^n}$⁠, the subgroup of alllinear combinations withinteger 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 aregular tiling of a space by aprimitive cell.

Lattices have many significant applications in pure mathematics, particularly in connection toLie algebras,number theory andgroup theory. They also arise in applied mathematics in connection withcoding theory, inpercolation theory to study connectivity arising from small-scale interactions,cryptography because of conjectured computational hardness of severallattice problems, and are used in various ways in the physical sciences. For instance, inmaterials science andsolid-state physics, a lattice is a synonym for the framework of acrystalline structure, a 3-dimensional array of regularly spaced points coinciding in special cases with theatom ormolecule positions in acrystal. More generally,lattice models are studied inphysics, often by the techniques ofcomputational physics.

A lattice is thesymmetry group of discretetranslational symmetry inn directions. A pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.^[3] As a group (dropping its geometric structure) a lattice is afinitely generatedfree abelian group, and thus isomorphic to⁠${\displaystyle {\mathbb{Z}}^n}$⁠.

A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. theatom ormolecule positions in acrystal, or more generally, the orbit of agroup 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${\displaystyle {\mathbb{R}}^n}$ is the subgroup⁠${\displaystyle {\mathbb{Z}}^n}$⁠. More complicated examples include theE₈ lattice, which is a lattice in⁠${\displaystyle {\mathbb{R}}^8}$⁠, and theLeech lattice in⁠${\displaystyle {\mathbb{R}}^{24}}$⁠. Theperiod lattice in${\displaystyle {\mathbb{R}}^2}$ is central to the study ofelliptic functions, developed in nineteenth century mathematics; it generalizes to higher dimensions in the theory ofabelian functions. Lattices calledroot lattices are important in the theory ofsimple Lie algebras; for example, the E₈ lattice is related to a Lie algebra that goes by the same name.

A lattice${\displaystyle \mathrm{\Lambda }}$ in${\displaystyle {\mathbb{R}}^n}$ thus has the form

${\displaystyle \mathrm{\Lambda }=\{\sum _{i=1}^n{a}_i{v}_i|a_i\in \mathbb{Z}\},}$

where$\{v_1,v_2,\dots ,v_n\}$ is a basis for⁠${\displaystyle {\mathbb{R}}^n}$⁠. Different bases can generate the same lattice, but thesquare root of thedeterminant of theGram matrix of the vectors$v_i$ is uniquely determined by${\displaystyle \mathrm{\Lambda }}$ and denoted by⁠${\displaystyle \mathrm{d}(\mathrm{\Lambda })}$⁠. If one thinks of a lattice as dividing the whole of${\displaystyle {\mathbb{R}}^n}$ into equalpolyhedra (copies of ann-dimensionalparallelepiped, known as thefundamental region of the lattice), then⁠${\displaystyle \mathrm{d}(\mathrm{\Lambda })}$⁠ is equal to then-dimensionalvolume of this polyhedron. This is why⁠${\displaystyle \mathrm{d}(\mathrm{\Lambda })}$⁠ is sometimes called thecovolume of the lattice. If this equals 1, the lattice is calledunimodular.

Minkowski's theorem relates the number⁠${\displaystyle \mathrm{d}(\mathrm{\Lambda })}$⁠ and the volume of a symmetricconvex set${\displaystyle S}$ to the number of lattice points contained in⁠${\displaystyle S}$⁠. The number of lattice points contained in apolytope all of whose vertices are elements of the lattice is described by the polytope'sEhrhart polynomial. Formulas for some of the coefficients of this polynomial involve⁠${\displaystyle \mathrm{d}(\mathrm{\Lambda })}$⁠ as well.

Computational lattice problems have many applications in computer science. For example, theLenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL) has been used in thecryptanalysis of manypublic-key encryption schemes,^[4] and manylattice-based cryptographic schemes are known to be secure under the assumption that certain lattice problems arecomputationally difficult.^[5]

There are five 2D lattice types as given by thecrystallographic restriction theorem. Below, thewallpaper group of the lattice is given inIUCr notation,Orbifold notation, andCoxeter notation, along with a wallpaper diagram showing the symmetry domains. Note that a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. Afull list of subgroups is available. For example, below the hexagonal/triangular lattice is given twice, with full 6-fold and a half 3-fold reflectional symmetry. If the symmetry group of a pattern contains ann-fold rotation then the lattice hasn-fold symmetry for evenn and 2n-fold for oddn.

cmm, (2*22), [∞,2+,∞]	p4m, (*442), [4,4]	p6m, (*632), [6,3]
rhombic lattice alsocenteredrectangular lattice isosceles triangular	square lattice right isosceles triangular	hexagonal lattice (equilateral triangular lattice)
pmm, *2222, [∞,2,∞]	p2, 2222, [∞,2,∞]+	p3m1, (*333), [3[3]]
rectangular lattice alsocentered rhombic lattice right triangular	oblique lattice scalene triangular	equilateraltriangular lattice (hexagonal lattice)

For the classification of a given lattice, start with one point and take a nearest second point. For the third point, not on the same line, consider its distances to both points. Among the points for which the smaller of these two distances is least, choose a point for which the larger of the two is least. (Notlogically equivalent but in the case of lattices giving the same result is just "Choose a point for which the larger of the two is least".)

The five cases correspond to thetriangle being equilateral, right isosceles, right, isosceles, and scalene. In a rhombic lattice, the shortest distance may either be a diagonal or a side of the rhombus, i.e., the line segment connecting the first two points may or may not be one of the equal sides of the isosceles triangle. This depends on the smaller angle of the rhombus being less than 60° or between 60° and 90°.

The general case is known as aperiod lattice. If the vectorsp andq generate the lattice, instead ofp andq we can also takep andp −q, etc. In general in 2D, we can takeap +bq andcp +dq for integersa,b,c andd such thatad-bc is 1 or −1. This ensures thatp andq themselves are integer linear combinations of the other two vectors. Each pairp,q defines a parallelogram, all with the same area, the magnitude of thecross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is afundamental parallelogram.

The vectorsp andq can be represented bycomplex numbers. Up to size and orientation, a pair can be represented by their quotient. Expressed geometrically: if two lattice points are 0 and 1, we consider the position of a third lattice point. Equivalence in the sense of generating the same lattice is represented by themodular group:${\displaystyle T:z\mapsto z+1}$ represents choosing a different third point in the same grid,${\displaystyle S:z\mapsto -1/z}$ represents choosing a different side of the triangle as reference side 0–1, which in general implies changing the scaling of the lattice, and rotating it. Each "curved triangle" in the image contains for each 2D lattice shape one complex number, the grey area is a canonical representation, corresponding to the classification above, with 0 and 1 two lattice points that are closest to each other; duplication is avoided by including only half of the boundary. The rhombic lattices are represented by the points on its boundary, with the hexagonal lattice as vertex, andi for the square lattice. The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammatic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis.

The 14 lattice types in 3D are calledBravais lattices. They are characterized by theirspace 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${\displaystyle {\mathbb{C}}^n}$ is a discrete subgroup of${\displaystyle {\mathbb{C}}^n}$ which spans${\displaystyle {\mathbb{C}}^n}$ as a real vector space. As the dimension of${\displaystyle {\mathbb{C}}^n}$ as a real vector space is equal to⁠${\displaystyle 2n}$⁠, a lattice in${\displaystyle {\mathbb{C}}^n}$ will be a free abelian group of rank⁠${\displaystyle 2n}$⁠.

For example, theGaussian integers${\displaystyle \mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}}$ form a lattice in⁠${\displaystyle \mathbb{C}={\mathbb{C}}^1}$⁠, as${\displaystyle (1,i)}$ is a basis of${\displaystyle \mathbb{C}}$ over⁠${\displaystyle \mathbb{R}}$⁠.

More generally, alattice Γ in aLie groupG is adiscrete subgroup, such that thequotientG/Γ is of finite measure, for the measure on it inherited fromHaar measure onG (left-invariant, or right-invariant—the definition is independent of that choice). That will certainly be the case whenG/Γ iscompact, but that sufficient condition is not necessary, as is shown by the case of themodular group inSL₂(R), which is a lattice but where the quotient isn't compact (it hascusps). There are general results stating the existence of lattices in Lie groups.

A lattice is said to beuniform orcocompact ifG/Γ is compact; otherwise the lattice is callednon-uniform.

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While we normally consider${\displaystyle \mathbb{Z}}$ lattices in${\displaystyle {\mathbb{R}}^n}$ this concept can be generalized to any finite-dimensionalvector space over anyfield. This can be done as follows:

LetK be afield, letV be ann-dimensionalK-vector space, let${\displaystyle B=\{{\mathbf{v}}_1,\dots ,{\mathbf{v}}_n\}}$ be aK-basis forV and letR be aring contained withinK. Then theR lattice${\displaystyle \mathcal{L}}$ inV generated byB is given by:

${\displaystyle \mathcal{L}=\{\sum _{i=1}^n{a}_i{\mathbf{v}}_i|a_i\in R\}.}$

In general, different basesB will generate different lattices. However, if thetransition matrix${\displaystyle T}$ between the bases is in${\displaystyle {\mathrm{G}\mathrm{L}}_n(R)}$ – thegeneral linear group of${\displaystyle R}$ (in simple terms this means that all the entries of${\displaystyle T}$ are in${\displaystyle R}$ and all the entries of${\displaystyle T^{-1}}$ are in${\displaystyle R}$ – which is equivalent to saying that thedeterminant ofT is in${\displaystyle R^{\ast }}$ – theunit group of elements inR with multiplicative inverses) then the lattices generated by these bases will beisomorphic sinceT induces an isomorphism between the two lattices.

Important cases of such lattices occur in number theory withK ap-adic field and${\displaystyle T}$ thep-adic integers.

For a vector space which is also aninner product space, thedual lattice can be concretely described by the set

${\displaystyle {\mathcal{L}}^{\ast }=\{\mathbf{v}\in V\mid ⟨\mathbf{v},\mathbf{x}⟩\in R\text{ for all }\mathbf{x}\in \mathcal{L}\},}$

or equivalently as

${\displaystyle {\mathcal{L}}^{\ast }=\{\mathbf{v}\in V\mid ⟨\mathbf{v},{\mathbf{v}}_i⟩\in R,i=1,...,n\}.}$

• Aprimitive element of a lattice is an element that is not a positive integer multiple of another element in the lattice.^{[citation needed]}

• Conway, John Horton;Sloane, Neil J. A. (1999),Sphere Packings, Lattices and Groups, Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York:Springer-Verlag,ISBN 978-0-387-98585-5,MR 0920369

• Catalogue of Lattices (by Nebe and Sloane)

• Lattice points
• Discrete groups
• Lie groups
• Analytic geometry

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