Sublattice, Covolume, Discriminant

Index Equals Lattice CountFor an integer lattice in real space, the index equals the lattice count.Apply a linear transformation if necessary, so that the containing lattice is the standard lattice in n space, i.e. the points with integer coordinates. Now each point is associated with a cube in space, a region of volume one. At the same time, the sublattice is spanned by bectors t1 through tn in n space, having integer coordinates. Since points and volume go hand in hand, intuition suggests the number of lattice points in the base cell equals the volume of the base cell, equals the determinant of the vectors t1 through tn that generate the parallelatope. We can prove this by looking at ever larger regions of space.

Source: Sublattice, Covolume, Discriminant