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

#### Comments

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.

#### References

[a1] | J.W.S. Cassels, "An introduction to the geometry of numbers" , Springer (1972) |

[a2] | P. Erdös, P.M. Gruber, J. Hammer, "Lattice points" , Longman (1989) |

[a3] | P.M. Gruber, C.G. Lekkerkerker, "Geometry of numbers" , North-Holland (1987) pp. Sect. (iv) (Updated reprint) |

[a4] | P.M. Gruber (ed.) J.M. Wills (ed.) , Handbook of convex geometry , North-Holland (1992) |

[a5] | R. Kannan, L. Lovasz, "Covering minima and lattice-point-free convex bodies" Ann. of Math. , 128 (1988) pp. 577–602 |

**How to Cite This Entry:**

Lattice of points.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lattice_of_points&oldid=17812