Integer lattices

In two dimensions, the integer lattice is the set of all points $(x, y)$ in the plane where $x$ and $y$ are integers. This can be written as $\mathbb{Z}^2$. In three dimensions, it is $\mathbb{Z}^3$, and so on for higher dimensions.

Important properties

• Lattice points are evenly spaced in all directions.

• The distance between two lattice points is always the distance between two pairs of integers.

• Lattice points are used to count, tile, and analyze geometric shapes in combinatorial geometry.

• The set $\mathbb{Z}^n$ is closed under addition and subtraction.