A regularly spaced array of points in a square array, i.e., points with coordinates , where , ,
... are integers. Such an array is often called a grid
or mesh, and is a special case of a point lattice.

The fraction of lattice points visible from the origin, as derived in Castellanos (1988, pp. 155-156),
is

(1)

(2)

(3)

Therefore, this is also the probability that two randomly picked integers will be
relatively prime to one another.

The number of the
lattice points
which can be picked with no four concyclic is (Guy 1994, p. 241).

Any parallelogram on the lattice in which two opposite
sides each have length 1 has unit area (Hilbert and Cohn-Vossen 1999, pp. 33-34).

A special set of polygons defined on the regular lattice are the golygons. A necessary
and sufficient condition that a linear transformation
transforms a lattice to itself is that it be unimodular.
M. Ajtai has shown that there is no efficient algorithm
for finding any fraction of a set of spanning vectors in a lattice having the shortest
lengths unless there is an efficient algorithm for all of them (of which none is
known). This result has potential applications to cryptography and authentication
(Cipra 1996).