Lattice of points
point lattice, in , with basis
The set of points , where are integers.
The lattice can be regarded as the free Abelian group with generators. A lattice has an infinite number of bases; their general form is , where runs through all integral matrices of determinant . The quantity
is the volume of the parallelopipedon formed by the vectors . It does not depend on the choice of a basis and is called the determinant of the lattice .
The idea of lattices and lattice points links geometry to arithmetic (integers). Therefore it plays a central role in the geometry of numbers; integer programming (lattice point theorems); Diophantine approximations; reduction theory; analytic number theory; numerical analysis; crystallography (cf. Crystallography, mathematical); coding and decoding; combinatorics; geometric algorithms, and other areas.
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Lattice of points. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lattice_of_points&oldid=24096