Three-dimensional periodicity and inversion axes in crystals

One of the basic tenets of the `lattice' is the strict adherence to 1-, 2-, 3-, 4- or 6-fold symmetry. The proof is simple and found in most crystallography text-books. However, proofs for inversion axes, which are distinct from rotation axes, are not available. The following presentation is not necessarily unique but may be unfamiliar to crystallographers and so is offered here as part of the new section on Teaching and Education in Crystallography.