Integer lattice
whose lattice points are n-tuples of integers.
The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2n n!.
As a matrix group it is given by the set of all n × n signed permutation matrices.
This group is isomorphic to the semidirect product where the symmetric group Sn acts on (Z2)n by permutation (this is a classic example of a wreath product).
In the study of Diophantine geometry, the square lattice of points with integer coordinates is often referred to as the Diophantine plane.
In mathematical terms, the Diophantine plane is the Cartesian product
The study of Diophantine figures focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integers.
In coarse geometry, the integer lattice is coarsely equivalent to Euclidean space.
Pick's theorem, first described by Georg Alexander Pick in 1899, provides a formula for the area of a simple polygon with all vertices lying on the 2-dimensional integer lattice, in terms of the number of integer points within it and on its boundary.
be the number of integer points interior to the polygon, and let
