Lattice

A lattice is a regular arrangement of points that extends through space with translational symmetry. In mathematics, a lattice in n-dimensional Euclidean space is a discrete subgroup of R^n that spans the real vector space. Equivalently, it consists of all integer linear combinations of a set of basis vectors b1, …, bn. The matrix with these basis vectors as columns defines the lattice. The volume of the fundamental parallelepiped formed by the basis is the determinant of the lattice, which measures its density. The dual lattice comprises all vectors that have integer inner products with every lattice vector. Lattices can be transformed by unimodular changes of basis, and their bases can be made shorter and closer to orthogonal through lattice reduction, such as the LLL algorithm. Lattice theory intersects number theory, geometry of numbers, and cryptography, with problems like shortest vector and closest vector being central.

In crystallography, a crystal lattice is an infinite periodic array of points representing atomic positions. The lattice is described by primitive vectors that generate a Bravais lattice, of which there are 14 distinct three-dimensional types. The unit cell is a repeating motif defined by lattice parameters (edge lengths and angles). The reciprocal lattice, important for diffraction, is used to describe periodicities in momentum space, and Miller indices label lattice planes.

In order theory, a lattice is a partially ordered set in which any two elements have a least upper bound (join) and a greatest lower bound (meet).

Applications of lattices span cryptography, computational number theory, solid-state physics, and computer graphics.