Integer lattices (other)

An integer lattice is a set of points in space where each point has integer coordinates. Beyond the standard grid (like all points $(x, y)$ where $x$ and $y$ are integers), we can consider more general lattices formed by integer combinations of vectors that are not necessarily aligned with the axes. These are called 'other' integer lattices.

Important properties

• A lattice can be generated by two or more linearly independent vectors with integer coordinates.

• Every point in the lattice is a sum of integer multiples of the generating vectors.

• The shape and orientation of the lattice depend on the choice of generating vectors.

• The area (in 2D) or volume (in higher dimensions) of the fundamental parallelogram/parallelepiped is related to the determinant of the matrix formed by the generating vectors.