Lattice of points

point lattice, in $ \mathbf R ^ {n} $, with basis $ [ \mathbf e _ {1} \dots \mathbf e _ {n} ] $

The set $ \Lambda = \mathbf Z {\mathbf e _ {1} } + \dots + \mathbf Z {\mathbf e _ {n} } $ of points $ \mathbf a = g _ {1} \mathbf e _ {1} + \dots + g _ {n} \mathbf e _ {n} $, where $ g _ {1} \dots g _ {n} $ are integers.

The lattice $ \Lambda $ can be regarded as the free Abelian group with $ n $ generators. A lattice $ \Lambda $ has an infinite number of bases; their general form is $ ( \mathbf e _ {1} \dots \mathbf e _ {n} ) U $, where $ U $ runs through all integral matrices of determinant $ \pm 1 $. The quantity

$$ d ( \Lambda ) = | \mathop{\rm det} ( \mathbf e _ {1} \dots e _ {n} ) | > 0 $$

is the volume of the parallelopipedon formed by the vectors $ \mathbf e _ {1} \dots \mathbf e _ {n} $. It does not depend on the choice of a basis and is called the determinant of the lattice $ \Lambda $.

The partition of point lattices into Voronoi lattice types plays an important role in the geometry of quadratic forms (cf. Quadratic form).

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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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