Editing Lattice (group) (section)

{{Short description|Periodic set of points}}
{{hatnote|Not to be confused with the partially ordered set, [[Lattice (order)]]. For other related uses, see [[Lattice (disambiguation)]].}}
{{One source|date=October 2022}}
[[File:Equilateral Triangle Lattice.svg|thumb|right|250px|A lattice in the [[Euclidean plane]]]]
{{Group theory sidebar |Discrete}}

In [[geometry]] and [[group theory]], a '''lattice''' in the [[real coordinate space]] <math>\mathbb{R}^n</math> 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]] 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]]. More abstractly, a lattice can be described as a [[free abelian group]] of dimension <math>n</math> which [[linear span|spans]] the [[vector space]] <math>\mathbb{R}^n</math>. For any [[basis (linear algebra)|basis]] of <math>\mathbb{R}^n</math>, the subgroup of all [[linear combination]]s with [[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]] of a space by a [[primitive cell]].

Lattices have many significant applications in pure mathematics, particularly in connection to [[Lie algebra]]s, [[number theory]] and [[group theory]]. They also arise in applied mathematics in connection with [[coding theory]], in [[percolation theory]] to study connectivity arising from small-scale interactions, [[cryptography]] because of conjectured computational hardness of several [[lattice problems]], and are used in various ways in the physical sciences. For instance, in [[materials science]] and [[solid-state physics]], a lattice is a synonym for the framework of a [[crystalline structure]], a 3-dimensional array of regularly spaced points coinciding in special cases with the [[atom]] or [[molecule]] positions in a [[crystal]]. More generally, [[lattice model (physics)|lattice models]] are studied in [[physics]], often by the techniques of [[computational physics]].