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Encyclopedia > Lattice group

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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Table of Geometry, from the 1728 Cyclopaedia. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In the mathematical subfield of linear algebra, the linear span of a set of vectors is the set of all linear combinations of the vectors. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ... In mathematics, linear combinations are a concept central to linear algebra and related fields of mathematics. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...


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, and are used in various ways in the physical sciences; for instance 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. In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... Coding theory is a branch of mathematics and computer science dealing with the error-prone process of transmitting data across noisy channels, via clever means, so that a large number of errors that occur can be corrected. ... To meet Wikipedias quality standards, this article or section may require cleanup. ... Properties In chemistry and physics, an atom (Greek άτομον meaning indivisible) is the smallest possible particle of a chemical element that retains its chemical properties. ... In chemistry, a molecule is an aggregate of at least two atoms in a definite arrangement held together by special forces. ... Quartz crystal In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ...


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. In physics, a lattice model is a physical model that is not defined on a continuum, but on a lattice, which is a graph or an n-complex approximating spacetime or space. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ...

Contents


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. The symmetry group of an object (e. ... A translation slides an object by a vector a: Ta(p) = p + a. ...


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 translate of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in the previous sense. 2-dimensional renderings (ie. ... Properties In chemistry and physics, an atom (Greek άτομον meaning indivisible) is the smallest possible particle of a chemical element that retains its chemical properties. ... In chemistry, a molecule is an aggregate of at least two atoms in a definite arrangement held together by special forces. ... Quartz crystal In chemistry and mineralogy, a crystal is a solid in which the constituent atoms, molecules, or ions are packed in a regularly ordered, repeating pattern extending in all three spatial dimensions. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G...


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. In mathematics, the Leech lattice is a lattice Λ in R24 discovered John Leech ( 16 (1964), 657--682). ... In mathematics, a fundamental pair of periods is an ordered pair of complex numbers that define a lattice in the complex plane. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i. ...


Dividing space according to a lattice

A typical lattice Λ 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 Λ, and is denoted by d(Λ). 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. In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... This article is about the geometric shape. ... In geometry, a parallelepiped (pronounced ; meaning of parallel planes) or parallelopipedon is a three-dimensional figure like a cube, except that its faces are not squares but parallelograms. ... In mathematics, given a lattice Γ in a Lie group G, a fundamental domain is a set D of representatives for the cosets G/Γ, that is also a well-behaved set topologically, in a sense that can be made precise in one of several ways. ... GEE GUY dimensions is called content. ...


Lattice points in convex sets

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. In mathematics, Minkowskis 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. ... Look up Convex set in Wiktionary, the free dictionary. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... In mathematics, integral polytopes have associated Ehrhart polynomials which encode some geometrical information about them. ...


Computing with lattices

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. The Lenstra-Lenstra-Lovász lattice reduction (LLL) algorithm is an algorithm which, given a lattice basis as input, outputs a basis with short, nearly orthogonal vectors. ... In computational complexity theory, polynomial time refers to the computation time of a problem where the time, m(n), is no greater than a polynomial function of the problem size, n. ... Public key cryptography is a form of cryptography which generally allows users to communicate securely without having prior access to a shared secret key. ...


Lattices in two dimensions: detailed discussion

There are five 2D lattice types as given by the crystallographic restriction theorem. Below the wallpaper group of the lattice is given in parentheses; note that a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself. If the symmetry group of a pattern contains an n-fold rotation then the lattice has n-fold symmetry for even n and 2n-fold for odd n. The crystallographic restriction theorem in its basic form is the observation that the rotational symmetries of a crystal are limited to 2-fold, 3-fold, 4-fold, and 6-fold. ... Example of an Egyptian design with wallpaper group p4m A wallpaper group (or plane crystallographic group) is a mathematical concept to classify repetitive designs on two-dimensional surfaces, such as walls, based on the symmetries in the pattern. ...

  • a rhombic lattice, also called centered rectangular lattice or isosceles triangular lattice (cmm), with evenly spaced rows of evenly spaced points, with the rows alternatingly shifted one half spacing (symmetrically staggered rows); special cases are:
    • a hexagonal lattice or equilateral triangular lattice (p6m)
    • a square lattice (see below, and turn 45°)
 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * 
 * * * * * * * * * * * * * * * * * * * * * * * * 
  • more generally, a parallelogrammic lattice, also called oblique lattice (p2)(with asymmetrically staggered rows):
 * * * * * * * * * * * * * * * * * * * * * * * * * * * * 

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. (Not logically 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".) This shape is a rhombus In geometry, a rhombus (also known as a rhomb) is a quadrilateral in which all of the sides are of equal length. ... A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ... Triangular tiling. ... In geometry, a rectangle is defined as a quadrilateral polygon in which all four angles are right angles. ... Upright square tiling. ... A parallelogram. ...


The five cases correspond to the triangle 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°. A triangle is one of the basic shapes of geometry: a polygon with three vertices and three sides which are straight line segments. ...


If the vectors a and b generate the lattice, instead of a and b we can also take a and a-b, etc. In general in 2D, we can take pa + qb and ra + sb for integers p, q, r, and s such that ps-qr is 1 or -1. This ensures that a and b themselves are integer linear combinations of the other two vectors. If not, not all translations are possible with the other pair. Each pair a, b defines a parallelogram, all with the same area, the magnitude of the cross product. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental domain. In mathematics, the cross product is a binary operation on vectors in a three-dimensional Euclidean space. ...

Enlarge

The vectors a and b can be represented by complex 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 the modular group: represents choosing a different third point in the same grid, 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, and i for the square lattice. The rectangular lattices are at the imaginary axis, and the remaining area represents the parallelogrammetic lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis. Image File history File links Not the best picture, but its the only one I have right now. ... Image File history File links Not the best picture, but its the only one I have right now. ... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...


Lattices in three dimensions

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. In geometry and crystallography, a Bravais lattice is an infinite set of points generated by a set of discrete translation operations. ... The space group of a crystal is a mathematical description of the symmetry inherent in the structure. ...


Lattices in complex space

A lattice in Cn is a discrete subgroup of Cn which spans the 2n-dimensional real vector space Cn. For example, the Gaussian integers form a lattice in C. A Gaussian integer is a complex number whose real and imaginary part are both integers. ...


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


In Lie groups

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. In mathematics, a Lie group is a group whose elements can be continuously parametrized by real numbers, such as the rotation group, which can be parametrized by the Euler angles. ... In mathematics, a discrete group is a group G equipped with the discrete topology. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is intuitively a group that collapses the normal subgroup N to the identity element. ... In mathematical analysis, the Haar measure is a way to assign an invariant volume to subsets of locally compact topological groups and subsequently define an integral for functions on those groups. ... Compact as a general noun can refer to: Look up Compact on Wiktionary, the free dictionary a diplomatic contract or covenant among parties, sometimes known as a pact, treaty, or an interstate compact; a British term for a newspaper format; In mathematics, it can refer to various concepts: Mostly commonly... In mathematics, the modular group Γ (Gamma) is a group that is a fundamental object of study in number theory, geometry, algebra, and many other areas of advanced mathematics. ...


A lattice is said to be uniform or cocompact if G/Γ is compact; otherwise the lattice is called non-uniform.


Lattices over general vector-spaces

Whilst we normally consider lattices in this concept can be generalised to any finite dimensional vector space over any field. This can be done as follows: In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... This article presents the essential definitions. ...


Let K be a field, let V be an n-dimensional K-vector space, let be a K-basis for V and let R be a ring contained within K. Then the R lattice in V generated by B is given by: This article presents the essential definitions. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, a basis or set of generators is a collection of objects that can be systematically combined to produce a larger collection of objects. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ...

Different bases B will in general generate different lattices. However, if the transition matrix T between the bases is in GLn(R) - the general linear group of R (in simple terms this means that all the entries of T are in R and all the entries of T − 1 are in R - which is equivalent to saying that the determinant of T is in R * - the unit group of elements in R with multiplicative inverses) then the lattices generated by these bases will be isomorphic since T induces an isomorphism between the two lattices. In mathematics, a (discrete-time) Markov chain is a discrete-time stochastic process with the Markov property. ... In mathematics, the general linear group of degree n over a field F (such as R or C), written as GL(n, F), is the group of n×n invertible matrices with entries from F, with the group operation that of ordinary matrix multiplication. ... In algebra, a determinant is a function depending on n that associates a scalar det(A) to every n×n square matrix A. The fundamental geometric meaning of a determinant is as the scale factor for volume when A is regarded as a linear transformation. ... In mathematics, a unit in a (unital) ring R is an invertible element of R, i. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...


Important cases occur in number theory with K a p-adic field and R the p-adic integers. To meet Wikipedias quality standards, this article or section may require cleanup. ... The title given to this article is incorrect due to technical limitations. ... The title given to this article is incorrect due to technical limitations. ...


See also


  Results from FactBites:
 
Translational symmetry - Wikipedia, the free encyclopedia (870 words)
For a given object, the translations for which this applies form a group, the symmetry group of the object, or, if the object has more kinds of symmetry, a subgroup of the symmetry group.
Different bases of translation vectors generate the same lattice iff one is transformed into the other by a matrix of integer coefficients of which the absolute value of the determinant is 1.
For two given lattice points, equivalence of choices of a third point to generate a lattice shape is represented by the modular group, see lattice.
  More results at FactBites »

 
 

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