lattici

Lattici is the plural form of lattice, a term used in several disciplines to denote either a regular, repeating arrangement of points in space or an abstract algebraic structure with two binary operations. In mathematics, a lattice is an algebraic structure consisting of a set equipped with meet and join, which define a partial order and enable aggregation of elements. In geometry and number theory, a lattice can be realized as the set of all integer linear combinations of a finite basis of vectors in Euclidean space. Such a lattice is often described as a lattice in R^n. Lattice bases can be transformed to shorter, more nearly orthogonal bases by lattice reduction algorithms (for example, the LLL algorithm). Lattices also appear in order theory as distributive or modular lattices, Boolean lattices, and more generally as partially ordered sets in which any two elements have a greatest lower bound and a least upper bound.

In crystallography and materials science, a lattice refers to a periodic array of points in space that serves as the skeleton of a crystal. Attaching a motif, or basis, to each lattice point generates the crystal structure. Bravais lattices classify the distinct translational symmetries in three dimensions; there are 14 three-dimensional Bravais lattices. The lattice constants a, b, c and the angles α, β, γ determine the unit cell. The reciprocal lattice is used to interpret diffraction experiments and relates to the Fourier transform of the lattice. Real crystals may contain defects such as vacancies and dislocations that perturb the ideal lattice.

In physics and computation, lattice concepts enable simulations by discretizing spacetime or fields. Lattice gauge theories and lattice quantum chromodynamics replace continuous spacetime by a grid, allowing numerical study of interactions. The term lattice thus links notions of regularity, symmetry, and discrete modeling across multiple disciplines.