The Geometry Junkyard

Lattice Theory and Geometry of Numbers

Informally, a lattice is an infinite arrangement of points spaced with sufficient regularity that one can shift any point onto any other point by some symmetry of the arrangement. More formally, a lattice can be defined as a discrete subgroup of a finite-dimensional vector space (the subgroup is often required not to lie within any subspace of the vector space, which can be expressed formally by saying that the quotient of the space by the lattice is compact).

The simplest and most commonly-studied example of a lattice (the "integer grid") is formed by the points all Cartesian coordinates of which are integers. Other types of lattices arise in crystallography and in sphere packing, where they are used to describe the locations of atoms or spheres. Lattices are also particularly important in the theory of periodic tilings, since they describe the set of translational symmetries of a tiling.

From the Geometry Junkyard, computational and recreational geometry pointers.
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David Eppstein, Theory Group, ICS, UC Irvine.
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