We find exact 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 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. We test the method by doing theoretical structure determination on the Pf1 virus from the Protein Data Bank. The underlying mathematical idea is that the structure is the orbit of a group, and this group is a subgroup of the invariance group of the differential equations. Joint work with Dominik Juestel and Gero Friesecke. (Acta Crystallographica A72 and SIAM J. Appl Math).
