Integer point

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

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 2ⁿn!. 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 (Z₂)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.

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 ${\displaystyle \scriptstyle \mathbb {Z} \times \mathbb {Z} }$ of the ring of all integers ${\displaystyle \scriptstyle \mathbb {Z} }$. The study of Diophantine figures focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integer.

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