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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
and
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
, a lattice in the
real coordinate space In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
\mathbb^n 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 In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
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 In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, a ...
. More abstractly, a lattice can be described as a
free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a sub ...
of dimension n which spans the
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
\mathbb^n. For any
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
of \mathbb^n, the subgroup of all linear combinations with
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language o ...
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 Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his ''Harmonices Mundi'' (Latin: ''The Harmony of the World'', 1619). Notation of Eucli ...
of a space by a
primitive cell In geometry, biology, mineralogy and solid state physics, a unit cell is a repeating unit formed by the vectors spanning the points of a lattice. Despite its suggestive name, the unit cell (unlike a unit vector, for example) does not necessaril ...
. Lattices have many significant applications in pure mathematics, particularly in connection to
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s,
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathe ...
and group theory. They also arise in applied mathematics in connection with
coding theory Coding theory is the study of the properties of codes and their respective fitness for specific applications. Codes are used for data compression, cryptography, error detection and correction, data transmission and data storage. Codes are studied ...
, in
percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnecte ...
to study connectivity arising from small-scale interactions,
cryptography Cryptography, or cryptology (from grc, , translit=kryptós "hidden, secret"; and ''graphein'', "to write", or ''-logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of adve ...
because of conjectured computational hardness of several
lattice problems In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: Lat ...
, and are used in various ways in the physical sciences. For instance, in materials science and
solid-state physics Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the ...
, a lattice is a synonym for the "framework of a
crystalline structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric patterns t ...
, a 3-dimensional array of regularly spaced points coinciding in special cases with the atom or
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
positions in a crystal. More generally,
lattice models In mathematical physics, a lattice model is a mathematical model of a physical system that is defined on a lattice, as opposed to a continuum, such as the continuum of space or spacetime. Lattice models originally occurred in the context of co ...
are studied in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relat ...
, often by the techniques of
computational physics Computational physics is the study and implementation of numerical analysis to solve problems in physics for which a quantitative theory already exists. Historically, computational physics was the first application of modern computers in science, ...
.


Symmetry considerations and examples

A lattice is the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
of discrete
translational symmetry In geometry, to translate a geometric figure is to move it from one place to another without rotating it. A translation "slides" a thing by . In physics and mathematics, continuous translational symmetry is the invariance of a system of equati ...
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 In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a sub ...
, and thus isomorphic to \mathbb^n. A lattice in the sense of a 3-
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
al array of regularly spaced points coinciding with e.g. the atom or
molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bioch ...
positions in a crystal, or more generally, the orbit of a
group action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
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 \mathbb^n is the subgroup \mathbb^n. More complicated examples include the
E8 lattice In mathematics, the E lattice is a special lattice in R. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E root system. The normIn ...
, which is a lattice in \mathbb^, and the
Leech lattice In mathematics, the Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was discovered by . It may also have been discovered (but not published) by ...
in\mathbb^. The period lattice iit\mathbb^2 is central to the study of
elliptic functions In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those i ...
, developed in nineteenth century mathematics; it generalizes to higher dimensions in the theory of
abelian function In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functio ...
s. Lattices called
root lattice In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...
s are important in the theory of
simple Lie algebra In algebra, a simple Lie algebra is a Lie algebra that is non-abelian and contains no nonzero proper ideals. The classification of real simple Lie algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of si ...
s; for example, the E8 lattice is related to a Lie algebra that goes by the same name.


Dividing space according to a lattice

A typical lattice \Lambda in \mathbb^n thus has the form : \Lambda = \left\ where is a basis for \mathbb^n. Different bases can generate the same lattice, but the
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of the vectors ''v''''i'' is uniquely determined by Λ, and is denoted by d(Λ). If one thinks of a lattice as dividing the whole of \mathbb^n into equal
polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...
(copies of an ''n''-dimensional
parallelepiped In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term ''rhomboid'' is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square. In Euclidean ...
, known as the ''
fundamental region Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
'' of the lattice), then d(Λ) is equal to the ''n''-dimensional
volume Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The def ...
of this polyhedron. This is why d(Λ) is sometimes called the covolume of the lattice. If this equals 1, the lattice is called unimodular.


Lattice points in convex sets

Minkowski's theorem In mathematics, Minkowski's theorem is the statement that every convex set in \mathbb^n which is symmetric with respect to the origin and which has volume greater than 2^n contains a non-zero integer point (meaning a point in \Z^n that is not ...
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 In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an - ...
all of whose vertices are elements of the lattice is described by the polytope's
Ehrhart polynomial In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a high ...
. Formulas for some of the coefficients of this polynomial involve d(Λ) as well.


Computational lattice problems

Computational lattice problems have many applications in computer science. For example, the
Lenstra–Lenstra–Lovász lattice basis reduction algorithm The Lenstra–Lenstra–Lovász (LLL) lattice basis reduction algorithm is a polynomial time lattice reduction algorithm invented by Arjen Lenstra, Hendrik Lenstra and László Lovász in 1982. Given a basis \mathbf = \ with ''n''-dimensional int ...
(LLL) has been used in the
cryptanalysis Cryptanalysis (from the Greek ''kryptós'', "hidden", and ''analýein'', "to analyze") refers to the process of analyzing information systems in order to understand hidden aspects of the systems. Cryptanalysis is used to breach cryptographic sec ...
of many
public-key encryption Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
schemes, and many lattice-based cryptographic schemes are known to be secure under the assumption that certain lattice problems are computationally difficult.


Lattices in two dimensions: detailed discussion

There are five 2D lattice types as given by the
crystallographic restriction theorem The crystallographic restriction theorem in its basic form was based on the observation that the rotational symmetries of a crystal are usually limited to 2-fold, 3-fold, 4-fold, and 6-fold. However, quasicrystals can occur with other diffraction ...
. Below, the
wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformatio ...
of the lattice is given in
IUC notation The International Union of Crystallography (IUCr) is an organisation devoted to the international promotion and coordination of the science of crystallography. The IUCr is a member of the International Council for Science (ICSU). Objectives T ...
,
Orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...
, and Coxeter 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. A full 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 an ''n''-fold rotation then the lattice has ''n''-fold symmetry for even ''n'' and 2''n''-fold for odd ''n''. 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 Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from premises ...
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 the
triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-collinear ...
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 a period lattice. If the vectors p and q generate the lattice, instead of p and q we can also take p and p-q, etc. In general in 2D, we can take ''a'' p + ''b'' q and ''c'' p + ''d'' q for integers ''a'',''b'', ''c'' and ''d'' such that ''ad-bc'' is 1 or -1. This ensures that p and q themselves are integer linear combinations of the other two vectors. Each pair p, q defines a parallelogram, all with the same area, the magnitude of the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is d ...
. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental parallelogram. The vectors p and q 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 In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
: T: z\mapsto z+1 represents choosing a different third point in the same grid, 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, and ''i'' 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.


Lattices in three dimensions

The 14 lattice types in 3D are called Bravais lattices. They are characterized by their
space group In mathematics, physics and chemistry, a space group is the symmetry group of an object in space, usually in three dimensions. The elements of a space group (its symmetry operations) are the rigid transformations of an object that leave it unch ...
. 3D patterns with translational symmetry of a particular type cannot have more, but may have less symmetry than the lattice itself.


Lattices in complex space

A lattice in \mathbb^n is a discrete subgroup of \mathbb^n which spans \mathbb C^n as a real vector space. As the dimension of \mathbb C^n as a real vector space is equal to 2n, a lattice in \mathbb^n will be a free abelian group of rank 2n. For example, the
Gaussian integer In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s \mathbb Z = \mathbb Z + i\mathbb Z form a lattice in \mathbb C = \mathbb^1, as (1, i) is a basis of \mathbb C over \mathbb R.


In Lie groups

More generally, a lattice Γ in a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
''G'' is a
discrete subgroup In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
, such that the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
''G''/Γ is of finite measure, for the measure on it inherited from
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...
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 Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
, but that sufficient condition is not necessary, as is shown by the case of the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional ...
in ''SL''2(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.


Lattices in general vector spaces

While we normally consider \mathbb lattices in \mathbb^n this concept can be generalized to any finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
over any
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. This can be done as follows: Let ''K'' be a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, let ''V'' be an ''n''-dimensional ''K''-
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
, let B = \ be a ''K''-
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
for ''V'' and let ''R'' be a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
contained within ''K''. Then the ''R'' lattice \mathcal in ''V'' generated by ''B'' is given by: :\mathcal = \left\. In general, different bases ''B'' will generate different lattices. However, if the transition matrix ''T'' between the bases is in \mathrm_n(R) - the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of ''R'' (in simple terms this means that all the entries of ''T'' are in ''R'' and all the entries of T^ are in ''R'' - which is equivalent to saying that the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of ''T'' is in R^* - the
unit group In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element of a ring is a unit if there exists in such that vu = uv = 1, where is the multiplicative identity; the element is unique for this ...
of elements in ''R'' with multiplicative inverses) then the lattices generated by these bases will be
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
since ''T'' induces an isomorphism between the two lattices. Important cases of such lattices occur in number theory with ''K'' a ''p''-adic field and ''R'' the ''p''-adic integers. For a vector space which is also an
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, the
dual lattice In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space. In certain respects, the geometry of the dual lattice of a lattice L is the reciprocal of the geometry of L , a perspective which underlie ...
can be concretely described by the set :\mathcal^* = \, or equivalently as :\mathcal^* = \.


Related notions

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


See also

*
Lattice (order) A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bo ...
* Lattice (module) *
Reciprocal lattice In physics, the reciprocal lattice represents the Fourier transform of another lattice (group) (usually a Bravais lattice). In normal usage, the initial lattice (whose transform is represented by the reciprocal lattice) is a periodic spatial fu ...
*
Unimodular lattice In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundame ...
* Crystal system *
Mahler's compactness theorem In mathematics, Mahler's compactness theorem, proved by , is a foundational result on lattices in Euclidean space, characterising sets of lattices that are 'bounded' in a certain definite sense. Looked at another way, it explains the ways in whi ...
*
Lattice graph In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a latti ...
*
Lattice-based cryptography Lattice-based cryptography is the generic term for constructions of cryptographic primitives that involve lattices, either in the construction itself or in the security proof. Lattice-based constructions are currently important candidates for post ...


Notes


References

*


External links


Catalogue of Lattices (by Nebe and Sloane)
{{DEFAULTSORT:Lattice (Group) Discrete groups Lie groups Analytic geometry