# Lattice (group)

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Periodic set of points

Not to be confused with the partially ordered set, [Lattice (order)](/source/Lattice_(order)). For other related uses, see [Lattice (disambiguation)](/source/Lattice_(disambiguation)).

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A lattice in the [Euclidean plane](/source/Euclidean_plane)

Algebraic structure → Group theory Group theory Basic notions Subgroup Normal subgroup Group action Quotient group (Semi-)direct product Direct sum Free product Wreath product Group homomorphisms kernel image simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable Glossary of group theory List of group theory topics Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic Discrete groups Lattices Integers ( Z {\displaystyle \mathbb {Z} } ) Free group Modular groups PSL(2, Z {\displaystyle \mathbb {Z} } ) SL(2, Z {\displaystyle \mathbb {Z} } ) Arithmetic group Lattice Hyperbolic group Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞) Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve v t e

In [geometry](/source/Geometry) and [group theory](/source/Group_theory), a **lattice** in the [real coordinate space](/source/Real_coordinate_space) R n {\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 the [square lattice](/source/Square_lattice), which consists of all points ( a , b ) {\displaystyle (a,b)} in the plane whose coordinates are both [integers](/source/Integer), and its higher-dimensional analogues the [integer lattices](/source/Integer_lattice) Z n {\displaystyle \mathbb {Z} ^{n}} .

Closure under addition and subtraction means that a lattice must be a [subgroup](/source/Subgroup) 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 a [Delone set](/source/Delone_set).[2]

More abstractly, a lattice can be described as a [free abelian group](/source/Free_abelian_group) of dimension n {\displaystyle n} which [spans](/source/Linear_span) the [vector space](/source/Vector_space) ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠. For any [basis](/source/Basis_(linear_algebra)) of ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠, the subgroup of all [linear combinations](/source/Linear_combination) with [integer](/source/Integer) 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](/source/Tessellation) of a space by a [primitive cell](/source/Primitive_cell).

Lattices have many significant applications in pure mathematics, particularly in connection to [Lie algebras](/source/Lie_algebra), [number theory](/source/Number_theory) and [group theory](/source/Group_theory). They also arise in applied mathematics in connection with [coding theory](/source/Coding_theory), in [percolation theory](/source/Percolation_theory) to study connectivity arising from small-scale interactions, [cryptography](/source/Cryptography) because of conjectured computational hardness of several [lattice problems](/source/Lattice_problems), and occur frequently in the physical sciences. For instance, in [materials science](/source/Materials_science) and [solid-state physics](/source/Solid-state_physics), a lattice is a synonym for a [crystalline structure](/source/Crystalline_structure), a 3-dimensional array of regularly spaced points coinciding in special cases with the [atom](/source/Atom) or [molecule](/source/Molecule) positions in a [crystal](/source/Crystal). More generally, [lattice models](/source/Lattice_model_(physics)) are studied in [physics](/source/Physics), often by the techniques of [computational physics](/source/Computational_physics).

## Symmetry considerations and examples

A lattice is the [symmetry group](/source/Symmetry_group) of discrete [translational symmetry](/source/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.[3] As a group (dropping its geometric structure) a lattice is a [finitely generated](/source/Finitely_generated_abelian_group) [free abelian group](/source/Free_abelian_group), and thus isomorphic to ⁠ Z n {\displaystyle \mathbb {Z} ^{n}} ⁠.

A lattice in the sense of a 3-[dimensional](/source/Dimension) array of regularly spaced points coinciding with e.g. the [atom](/source/Atom) or [molecule](/source/Molecule) positions in a [crystal](/source/Crystal), or more generally, the orbit of a [group action](/source/Group_action_(mathematics)) 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 R n {\displaystyle \mathbb {R} ^{n}} is the subgroup ⁠ Z n {\displaystyle \mathbb {Z} ^{n}} ⁠. More complicated examples include the [E8 lattice](/source/E8_lattice), which is a lattice in ⁠ R 8 {\displaystyle \mathbb {R} ^{8}} ⁠, and the [Leech lattice](/source/Leech_lattice) in ⁠ R 24 {\displaystyle \mathbb {R} ^{24}} ⁠. The [period lattice](/source/Period_lattice) in R 2 {\displaystyle \mathbb {R} ^{2}} is central to the study of [elliptic functions](/source/Elliptic_functions), developed in nineteenth century mathematics; it generalizes to higher dimensions in the theory of [abelian functions](/source/Abelian_function). Lattices called [root lattices](/source/Root_lattice) are important in the theory of [simple Lie algebras](/source/Simple_Lie_algebra); for example, the E8 lattice is related to a Lie algebra that goes by the same name.

## Lattice basis tiling space

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

- Λ = { ∑ i = 1 n a i v i | a i ∈ Z } , {\displaystyle \Lambda ={\biggl \{}\sum _{i=1}^{n}a_{i}v_{i}\mathbin {\bigg \vert } a_{i}\in \mathbb {Z} {\biggr \}},}

where { v 1 , v 2 , … , v n } {\textstyle \{v_{1},v_{2},\ldots ,v_{n}\}} is a basis for ⁠ R n {\displaystyle \mathbb {R} ^{n}} ⁠, whose [column-vectors](/source/Row_and_column_vectors) form an *n* x *n* matrix *M*. Any other basis { v 1 ′ , v 2 ′ , … , v n ′ } {\textstyle \{v_{1}',v_{2}',\ldots ,v_{n}'\}} with matrix *M'* is related by an [automorphism](/source/Group_isomorphism) of the group Λ ≅ Z n {\displaystyle \Lambda \cong \mathbb {Z} ^{n}} , meaning M ′ = A M {\displaystyle M'=AM} for an integer transition matrix A ∈ G L n ( Z ) {\displaystyle A\in \mathrm {GL} _{n}(\mathbb {Z} )} having det ( A ) = ± 1 {\displaystyle \det(A)=\pm 1} .

A lattice fills the whole of R n {\displaystyle \mathbb {R} ^{n}} with equal [tiles](/source/Tessellation), copies of the *n*-dimensional [parallelepiped](/source/Parallelepiped) spanned by the basis vectors, known as the *[fundamental domain](/source/Fundamental_region)* or *primitive cell* of the lattice. The *n*-dimensional [volume](/source/Volume) of this fundamental domain is sometimes called the **covolume** of the lattice: it is invariant for any basis, and may be computed as d ( Λ ) = | det ( M ) | {\displaystyle d(\Lambda )=|\det(M)|} . (If the *n*-dimensional lattice *Λ {\displaystyle \Lambda }* lies in a higher-dimensional space R m {\displaystyle \mathbb {R} ^{m}} , its basis forms an *m* x *n* matrix whose volume can be computed using the [Gram matrix](/source/Gram_matrix): d ( Λ ) = det M t M {\displaystyle d(\Lambda )={\sqrt {\det M^{t}M}}} .) A lattice with d ( Λ ) = 1 {\displaystyle d(\Lambda )=1} is called [unimodular](/source/Unimodular_lattice).

## Lattice points in convex sets

[Minkowski's theorem](/source/Minkowski's_theorem) relates the number ⁠ d ( Λ ) {\displaystyle \mathrm {d} (\Lambda )} ⁠, or more generally the volume of a symmetric [convex set](/source/Convex_set) S {\displaystyle S} , to the number of lattice points contained in ⁠ S {\displaystyle S} ⁠. For a [polytope](/source/Polytope) whose vertices are elements of the lattice, the number of lattice points it contains is described by the polytope's [Ehrhart polynomial](/source/Ehrhart_polynomial). Formulas for some of the coefficients of this polynomial involve ⁠ d ( Λ ) {\displaystyle \mathrm {d} (\Lambda )} ⁠ as well.

See also: [Integer points in polyhedra](/source/Integer_points_in_polyhedra)

## Computational lattice problems

Main article: [Lattice problem](/source/Lattice_problem)

[Computational lattice problems](/source/Lattice_problems) have many applications in computer science. For example, the [Lenstra–Lenstra–Lovász lattice basis reduction algorithm](/source/Lenstra%E2%80%93Lenstra%E2%80%93Lov%C3%A1sz_lattice_basis_reduction_algorithm) (LLL) has been used in the [cryptanalysis](/source/Cryptanalysis) of many [public-key encryption](/source/Public-key_cryptography) schemes,[4] and many [lattice-based cryptographic schemes](/source/Lattice-based_cryptography) are known to be secure under the assumption that certain lattice problems are [computationally difficult](/source/Computational_hardness_assumption).[5]

## Lattices in two dimensions: detailed discussion

Five lattices in the Euclidean plane

There are five 2D lattice types as given by the [crystallographic restriction theorem](/source/Crystallographic_restriction_theorem). Below, the [wallpaper group](/source/Wallpaper_group) of the lattice Λ {\displaystyle \Lambda } is given in [IUCr notation](/source/IUC_notation), [Orbifold notation](/source/Orbifold_notation), and [Coxeter notation](/source/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](/source/List_of_planar_symmetry_groups#Wallpaper_groups) 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*.

cmm, (2*22), [∞,2+,∞] p4m, (*442), [4,4] p6m, (*632), [6,3] rhombic lattice also centered rectangular 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 also centered rhombic lattice right triangular oblique lattice scalene triangular equilateral triangular 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. (Not [logically equivalent](/source/Logical_equivalence) 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](/source/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°.

The general case is known as a [period lattice](/source/Period_lattice). The vectors {**p**,**q**} are a generator pair or a basis of the lattice Λ {\displaystyle \Lambda } . Instead of {**p**, **q**} we can also take the basis {**p**, **p** − **q**}, or in general {*a***p** + *b***q** **,** *c***p** + *d***q**} for integers *a*,*b*,*c*,*d* forming an integer transition matrix T = ( a b c d ) {\displaystyle T={\bigl (}{\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}{\bigr )}} of unit determinant, meaning det T = a d − b c = ± 1 {\displaystyle \det T=ad-bc=\pm 1} . This ensures that **p** and **q** themselves are integer linear combinations of the other two vectors. (The transition matrix T {\displaystyle T} lies in G L 2 ( Z ) {\displaystyle \mathrm {GL} _{2}(\mathbb {Z} )} , the automorphism group of the lattice L ≅ Z 2 {\displaystyle L\cong \mathbb {Z} ^{2}} , which is a double cover of the well-studied [modular group](/source/Modular_group) S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} .)

Each basis pair {**p**, **q**} defines a parallelogram, all with the same area given by the magnitude of the [cross product](/source/Cross_product) **p** x **q**. This parallelogram is a [fundamental parallelogram](/source/Fundamental_pair_of_periods) of the translation symmetries, i.e. a fundamental domain or primitive cell.

The [fundamental domain](/source/Fundamental_domain) of the [period lattice](/source/Period_lattice).

The basis vectors {**p,q**} can be represented by [complex numbers](/source/Complex_number). Up to changing the scaling of the lattice and rotating it, the pair {**p,q**} can be represented by their complex number quotient: if we fix two standard lattice points 0 and 1 in the complex plane, the lattice shape is determined by the third lattice point *z = p*/*q*. A change of basis is represented by the modular group S L 2 ( Z ) {\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )} , which acts on the complex plane by [linear fractional transformations](/source/Linear_fractional_transformation), generated by the two operations T : z ↦ z + 1 {\displaystyle T:z\mapsto z+1} , shifting to a different third point in the same grid, and S : z ↦ − 1 / z {\displaystyle S:z\mapsto -1/z} , choosing a different side of the triangle as reference side 0–1. The figure shows the action of the modular group acting on the complex plane C {\displaystyle \mathbb {C} } (not to be confused with the lattice translating the real plane R 2 {\displaystyle \mathbb {R} ^{2}} ). Each "curved triangle" in the image is a fundamental domain of the modular group, contain one complex number for each 2D lattice Λ {\displaystyle \Lambda } up to scaling and rotation. The grey area is a standard fundamental domain, corresponding to the canonical representation of Λ {\displaystyle \Lambda } with 0 and 1 being 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 the 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 parallelogram lattices, with the mirror image of a parallelogram represented by the mirror image in the imaginary axis.

## Lattices in three dimensions

Main article: [Bravais lattice](/source/Bravais_lattice)

The 14 lattice types in 3D are called **Bravais lattices**. They are characterized by their [space group](/source/Space_group). 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 C n {\displaystyle \mathbb {C} ^{n}} is a discrete subgroup of C n {\displaystyle \mathbb {C} ^{n}} which spans C n {\displaystyle \mathbb {C} ^{n}} as a real vector space. As the dimension of C n {\displaystyle \mathbb {C} ^{n}} as a real vector space is equal to ⁠ 2 n {\displaystyle 2n} ⁠, a lattice in C n {\displaystyle \mathbb {C} ^{n}} will be a free abelian group of rank ⁠ 2 n {\displaystyle 2n} ⁠.

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

## In Lie groups

Main article: [Lattice (discrete subgroup)](/source/Lattice_(discrete_subgroup))

More generally, a **lattice** Γ in a [Lie group](/source/Lie_group) *G* is a [discrete subgroup](/source/Discrete_subgroup), such that the [quotient](/source/Quotient_group) *G*/Γ is of finite measure, for the measure on it inherited from [Haar measure](/source/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](/source/Compact_space), but that sufficient condition is not necessary, as is shown by the case of the [modular group](/source/Modular_group) in [SL2(**R**)](/source/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.

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

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While we normally consider Z {\displaystyle \mathbb {Z} } lattices in R n {\displaystyle \mathbb {R} ^{n}} this concept can be generalized to any finite-dimensional [vector space](/source/Vector_space) over any [field](/source/Field_(mathematics)). This can be done as follows:

Let *K* be a [field](/source/Field_(mathematics)), let *V* be an *n*-dimensional *K*-[vector space](/source/Vector_space), let B = { v 1 , … , v n } {\displaystyle B=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}} be a *K*-[basis](/source/Basis_(linear_algebra)) for *V* and let *R* be a [ring](/source/Ring_(mathematics)) contained within *K*. Then the *R* lattice L {\displaystyle {\mathcal {L}}} in *V* generated by *B* is given by:

- L = { ∑ i = 1 n a i v i | a i ∈ R } . {\displaystyle {\mathcal {L}}={\biggl \{}\sum _{i=1}^{n}a_{i}\mathbf {v} _{i}\mathbin {\bigg \vert } a_{i}\in R{\biggr \}}.}

In general, different bases *B* will generate different lattices. However, if the [transition matrix](/source/Change_of_basis#General_case) T {\displaystyle T} between the bases is in G L n ( R ) {\displaystyle \mathrm {GL} _{n}(R)} – the [general linear group](/source/General_linear_group) of R {\displaystyle R} (in simple terms this means that all the entries of T {\displaystyle T} are in R {\displaystyle R} and all the entries of T − 1 {\displaystyle T^{-1}} are in R {\displaystyle R} – which is equivalent to saying that the [determinant](/source/Determinant) of *T* is in R ∗ {\displaystyle R^{*}} – the [unit group](/source/Unit_group) of elements in *R* with multiplicative inverses) then the lattices generated by these bases will be [isomorphic](/source/Isomorphism) since *T* induces an isomorphism between the two lattices.

Important cases of such lattices occur in number theory with *K* a [*p*-adic field](/source/P-adic_field) and T {\displaystyle T} the [*p*-adic integers](/source/P-adic_integers).

For a vector space which is also an [inner product space](/source/Inner_product_space), the [dual lattice](/source/Dual_lattice) can be concretely described by the set

- L ∗ = { v ∈ V ∣ ⟨ v , x ⟩ ∈ R for all x ∈ L } , {\displaystyle {\mathcal {L}}^{*}=\{\mathbf {v} \in V\mid \langle \mathbf {v} ,\mathbf {x} \rangle \in R\,{\text{ for all }}\,\mathbf {x} \in {\mathcal {L}}\},}

or equivalently as

- L ∗ = { v ∈ V ∣ ⟨ v , v i ⟩ ∈ R , i = 1 , . . . , n } . {\displaystyle {\mathcal {L}}^{*}=\{\mathbf {v} \in V\mid \langle \mathbf {v} ,\mathbf {v} _{i}\rangle \in R,\ i=1,...,n\}.}

## Saturated lattices

A **primitive element** of a lattice is an element v ∈ Λ {\displaystyle v\in \Lambda } that is not a positive integer multiple of another element in the lattice.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*] If Λ {\displaystyle \Lambda } has basis { v 1 , … , v n } {\displaystyle \{v_{1},\ldots ,v_{n}\}} , we can identify it with the standard lattice Λ ≅ Z n {\displaystyle \Lambda \cong \mathbb {Z} ^{n}} via a 1 v 1 + ⋯ a n v n ↦ ( a 1 , … , a n ) {\displaystyle a_{1}v_{1}+\cdots a_{n}v_{n}\mapsto (a_{1},\ldots ,a_{n})} ; then a vector v = ( a 1 , … , a n ) {\displaystyle v=(a_{1},\ldots ,a_{n})} is primitive whenever 1 d v ∉ Z n {\displaystyle {\tfrac {1}{d}}v\notin \mathbb {Z} ^{n}} for any integer d > 1 {\displaystyle d>1} , which is equivalent to the coordinates being [coprime](/source/Coprime_integers), gcd ( a 1 , … , a n ) = 1 {\displaystyle \gcd(a_{1},\ldots ,a_{n})=1} . Every one-dimensional sublattice Γ ⊂ Λ {\displaystyle \Gamma \subset \Lambda } has a primitive generator which is unique up to sign.

More generally, consider an ℓ {\displaystyle \ell } -dimensional sublattice Γ ⊂ Z n {\displaystyle \Gamma \subset \mathbb {Z} ^{n}} with a basis { v 1 , … , v ℓ } {\displaystyle \{v_{1},\ldots ,v_{\ell }\}} whose column vectors form a matrix M ∈ M n × ℓ ( Z ) {\displaystyle M\in \mathrm {M} _{n\times \ell }(\mathbb {Z} )} . We say Γ {\displaystyle \Gamma } is a **saturated sublattice** whenever any of the following equivalent conditions holds:

- Γ = V ∩ Z n {\displaystyle \Gamma =V\cap \mathbb {Z} ^{n}} for some ℓ {\displaystyle \ell } -dimensional Q {\displaystyle \mathbb {Q} } -[linear subspace](/source/Linear_subspace) V ⊂ Q n {\displaystyle V\subset \mathbb {Q} ^{n}}

- For every v ∈ Z n {\displaystyle v\in \mathbb {Z} ^{n}} and integer d > 1 {\displaystyle d>1} , we have d v ∈ Γ {\displaystyle dv\in \Gamma } only when v ∈ Γ {\displaystyle v\in \Gamma } .

- The quotient group Z n / Γ {\displaystyle \mathbb {Z} ^{n}/\Gamma } is a [free abelian group](/source/Free_abelian_group), without [torsion](/source/Torsion_(algebra)).

- Z n = Γ ⊕ Γ ⊥ {\displaystyle \mathbb {Z} ^{n}=\Gamma \oplus \Gamma ^{\perp }} , where Γ ⊥ {\displaystyle \Gamma ^{\perp }} is the orthogonal subspace in Z n {\displaystyle \mathbb {Z} ^{n}} with respect to the standard [dot product](/source/Dot_product).

- The basis matrix M ∈ M n × ℓ ( Z ) {\displaystyle M\in \mathrm {M} _{n\times \ell }(\mathbb {Z} )} possesses an integer left inverse N ∈ M ℓ × n ( Z ) {\displaystyle N\in \mathrm {M} _{\ell \times n}(\mathbb {Z} )} with N M = I d n {\displaystyle NM=\mathrm {Id} _{n}} .

- The [Smith normal form](/source/Smith_normal_form) of M {\displaystyle M} has only 1's on the main diagonal.

- The maximal [minors](/source/Minor_(linear_algebra)) of M {\displaystyle M} are coprime: gcd { Δ I ( M ) } = 1 {\displaystyle \gcd\{\Delta _{I}(M)\}=1} , where I {\displaystyle I} runs over all *n*-element subsets of { 1 , … , m } {\displaystyle \{1,\ldots ,m\}} .

## See also

- [Crystal system](/source/Crystal_system)

- [Hermite constant](/source/Hermite_constant)

- [Lattice-based cryptography](/source/Lattice-based_cryptography)

- [Lattice graph](/source/Lattice_graph)

- [Lattice (module)](/source/Lattice_(module))

- [Lattice (order)](/source/Lattice_(order))

- [Mahler's compactness theorem](/source/Mahler's_compactness_theorem)

- [Reciprocal lattice](/source/Reciprocal_lattice)

- [Unimodular lattice](/source/Unimodular_lattice)

## Notes

1. **[^](#cite_ref-1)** Gruber, Peter M.; Lekkerkerker, Cornelis G. (1987). *Geometry of numbers*. North-Holland Mathematical Library (Second ed.). Amsterdam, The Netherlands: North-Holland. [ISBN](/source/ISBN_(identifier)) [978-0-08-096023-4](https://en.wikipedia.org/wiki/Special:BookSources/978-0-08-096023-4).

1. **[^](#cite_ref-2)** Baake, Michael; Grimm, Uwe (2013). *Aperiodic order*. Encyclopedia of mathematics and its applications. Cambridge; New York: Cambridge University Press. [ISBN](/source/ISBN_(identifier)) [978-0-521-86991-1](https://en.wikipedia.org/wiki/Special:BookSources/978-0-521-86991-1).

1. **[^](#cite_ref-3)** ["Symmetry in Crystallography Notes"](https://web.archive.org/web/20220826095614/http://xrayweb.chem.ou.edu/notes/symmetry.html). *xrayweb.chem.ou.edu*. Archived from [the original](http://xrayweb.chem.ou.edu/notes/symmetry.html) on 2022-08-26. Retrieved 2022-11-06.

1. **[^](#cite_ref-4)** Nguyen, Phong; Stern, Jacques (2001). "The Two Faces of Lattices in Cryptology". *Cryptography and Lattices*. Lecture Notes in Computer Science. Vol. 2146. pp. 146–180. [doi](/source/Doi_(identifier)):[10.1007/3-540-44670-2_12](https://doi.org/10.1007%2F3-540-44670-2_12). [ISBN](/source/ISBN_(identifier)) [978-3-540-42488-8](https://en.wikipedia.org/wiki/Special:BookSources/978-3-540-42488-8).

1. **[^](#cite_ref-5)** Regev, Oded (2005-01-01). "On lattices, learning with errors, random linear codes, and cryptography". *Proceedings of the thirty-seventh annual ACM symposium on Theory of computing*. STOC '05. New York, NY, USA: ACM. pp. 84–93. [CiteSeerX](/source/CiteSeerX_(identifier)) [10.1.1.110.4776](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.4776). [doi](/source/Doi_(identifier)):[10.1145/1060590.1060603](https://doi.org/10.1145%2F1060590.1060603). [ISBN](/source/ISBN_(identifier)) [978-1581139600](https://en.wikipedia.org/wiki/Special:BookSources/978-1581139600). [S2CID](/source/S2CID_(identifier)) [53223958](https://api.semanticscholar.org/CorpusID:53223958).

## References

- [Conway, John Horton](/source/John_Horton_Conway); [Sloane, Neil J. A.](/source/Neil_Sloane) (1999), [*Sphere Packings, Lattices and Groups*](/source/Sphere_Packings%2C_Lattices_and_Groups), Grundlehren der Mathematischen Wissenschaften, vol. 290 (3rd ed.), Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [ISBN](/source/ISBN_(identifier)) [978-0-387-98585-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98585-5), [MR](/source/MR_(identifier)) [0920369](https://mathscinet.ams.org/mathscinet-getitem?mr=0920369)

## External links

- [Catalogue of Lattices (by Nebe and Sloane)](http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/)

v t e Crystal systems Bravais lattice Crystallographic point group Seven 3D systems triclinic (anorthic) monoclinic orthorhombic tetragonal trigonal & hexagonal cubic (isometric) Four 2D systems oblique rectangular square hexagonal

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