Periodic Functions, Lattices and Their Projections

Functions whose symmetries form a crystallographic group in particular have a lattice of periods, and the set of their level curves forms a periodic pattern. We show how after projecting these functions, one obtains new functions with a lattice of periods that is not the projection of the initial lattice. We also characterise all the crystallographic groups in three dimensions that are symmetry groups of patterns whose projections have periods in a given two-dimensional lattice. The particular example of patterns that after projection have a hexagonal lattice of periods is discussed in detail.