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Inmathematics, then-dimensionalinteger lattice (orcubic lattice), denoted⁠${\displaystyle {\mathbb{Z}}^n}$⁠, is thelattice in theEuclidean space⁠${\displaystyle {\mathbb{R}}^n}$⁠ whose lattice points aren-tuples ofintegers. The two-dimensional integer lattice is also called thesquare lattice, or grid lattice.⁠${\displaystyle {\mathbb{Z}}^n}$⁠ is the simplest example of aroot lattice. The integer lattice is an oddunimodular lattice.

Theautomorphism group (orgroup ofcongruences) of the integer lattice consists of allpermutations and sign changes of the coordinates, and is oforder 2ⁿ n!. As amatrix group it is given by the set of alln × nsigned permutation matrices. This group isisomorphic to thesemidirect product

${\displaystyle ({\mathbb{Z}}_2{)}^n⋊S_n}$

where thesymmetric groupS_n acts on (Z₂)ⁿ by permutation (this is a classic example of awreath product).

For the square lattice, this is the group of thesquare, or thedihedral group of order 8; for the three-dimensional cubic lattice, we get the group of thecube, oroctahedral group, of order 48.

In the study ofDiophantine geometry, the square lattice of points with integer coordinates is often referred to as theDiophantine plane. In mathematical terms, the Diophantine plane is theCartesian product${\displaystyle \mathbb{Z}\times \mathbb{Z}}$ of thering of all integers${\displaystyle \mathbb{Z}}$. The study ofDiophantine figures focuses on the selection of nodes in the Diophantine plane such that all pairwise distances are integers.

Incoarse geometry, the integer lattice is coarsely equivalent toEuclidean space.

Pick's theorem, first described byGeorg Alexander Pick in 1899, provides a formula for thearea of asimple polygon with allvertices lying on the 2-dimensional integer lattice, in terms of the number of integer points within it and on its boundary.^[1]

Let${\displaystyle i}$ be the number of integer points interior to the polygon, and let${\displaystyle b}$ be the number of integer points on its boundary (including both vertices and points along the sides). Then the area${\displaystyle A}$ of this polygon is:^[2]$${\displaystyle A=i+\frac{b}{2}-1.}$$The example shown has${\displaystyle i=7}$ interior points and${\displaystyle b=8}$ boundary points, so its area is${\displaystyle A=7+\frac{8}{2}-1=10}$ square units.

• Regular grid

• Olds, C. D.;Lax, Anneli;Davidoff, Giuliana (2000).The Geometry of Numbers. New Mathematical Library. Vol. 41. Mathematical Association of America.ISBN 0-88385-643-3.

• Euclidean geometry
• Lattice points
• Diophantine geometry

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