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In
mathematics, the n-dimensional **integer lattice** (or **cubic lattice**), denoted , is the
lattice in the
Euclidean space whose lattice points are
n-tuples of
integers. The two-dimensional integer lattice is also called the
square lattice, or grid lattice. 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 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.

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.

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.^{
[1]}

Let be the number of integer points interior to the polygon, and let be the number of integer points on its boundary (including both vertices and points along the sides). Then the
area of this polygon is:^{
[2]}

The example shown has interior points and boundary points, so its area is square units.

- Olds, C.D. et al. (2000).
*The Geometry of Numbers*. Mathematical Association of America. ISBN 0-88385-643-3.`{{ cite book}}`

: CS1 maint: uses authors parameter ( link)

**^**Pick, Georg (1899). "Geometrisches zur Zahlenlehre".*Sitzungsberichte des deutschen naturwissenschaftlich-medicinischen Vereines für Böhmen "Lotos" in Prag*. (Neue Folge).**19**: 311–319. JFM 33.0216.01. CiteBank:47270**^**Aigner, Martin; Ziegler, Günter M. (2018). "Three applications of Euler's formula: Pick's theorem".*Proofs from THE BOOK*(6th ed.). Springer. pp. 93–94. doi: 10.1007/978-3-662-57265-8. ISBN 978-3-662-57265-8.