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, in 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 "frame work" 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 models are studied in physics, often by the techniques of computational physics.
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.
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.
Computational lattice problems have many applications in computer science. For example, the (LLL) has been used in the cryptanalysis of many public-key encryption schemes, and many lattice-based cryptographic schemes are known to be secure under the assumption that certain lattice problems are computationally difficult.
There are five 2D lattice types as given by the crystallographic restriction theorem. Below, the wallpaper group of the lattice is given in IUC notation, Orbifold notation, 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 2n-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 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 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. One parallelogram fully defines the whole object. Without further symmetry, this parallelogram is a fundamental parallelogram.
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.
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.
A lattice is said to be uniform or cocompact if G/Γ is compact; otherwise the lattice is called non-uniform.