In mathematics, the *n*-dimensional **integer lattice** (or **cubic lattice**), denoted **Z**^{n}, is the lattice in the Euclidean space **R**^{n} whose lattice points are *n*-tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. **Z**^{n} is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice.

## Automorphism group

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 2^{n} *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 *S*_{n} acts on (**Z**_{2})^{n} by permutation (this is a classic example of a wreath product).

For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, or octahedral group, of order 48.

## Diophantine geometry

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 of the ring of all integers . The study of Diophantine figures focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integer.

## Coarse geometry

In coarse geometry, the integer lattice is coarsely equivalent to Euclidean space.

## See also

## References

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- Olds, C.D. et al. (2000).
*The Geometry of Numbers*. Mathematical Association of America. ISBN 0-88385-643-3.CS1 maint: Uses authors parameter (link)