After a general introduction to our work on “objective structures”, we focus on Maxwell’s equations. We find solutions of Maxwell’s equations that are the precise analog of plane waves, but in the case that the translation group is replaced by the (largest) Abelian helical group. These waves display constructive/destructive interference with helical atomic structures, in the same way that plane waves interact with crystals. We show how the resulting far-field pattern can be used for structure determination, and we test the idea theoretically on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea of this and our related work is always the same: the structure of interest is the orbit of a group, and this group is an invariance group of the differential equations. Joint work with Dominik Juestel and Gero Friesecke (SIAM J. Applied Math 76 and Acta Cryst. A72).