On visualizing crystal lattice planes

Given the primitive vectors of an arbitrary Bravais lattice and the Miller indices of a set of lattice planes in it, it is shown how to construct an alternative set of primitive vectors that are adapted to the lattice planes in the following sense: all but one of these alternative vectors serve as a basis for the points in one of the lattice planes, and the remaining vector serves to shift any of these lattice planes into a neighboring one. This construction is described for a three-dimensional Bravais lattice and then generalized to arbitrary d-dimensional lattices. This construction can be used to generate computer images of lattice points in a succession of crystal lattice planes, which could be useful for instructional purposes.