UC Riverside

In crystallography, it was an axiom that any material with a diffraction pattern consisting of sharp spots must have an atomic structure that is a repetition of a unit cell consisting of finitely many atoms, i.e., having translational symmetry in three linearly independent spacial directions. However, the discovery of quasicrystals in 1982 by Dan Shechtman, which defied this law, started an investigation into the question of exactly which atomic structure is necessary to produce a pure point diffraction pattern. In this thesis, we give an extended version of the Poisson summation formula, and use it to address a question by Lapidus and van Frankenhuijsen on whether or not the complex dimensions of a nonlattice self-similar fractal string can be understood as a mathematical analog for a quasicrystal, in the context of a mathematical idealization of diffraction developed by A. Hof. Also, we provide an implementation of the LLL algorithm to give an exploration of the quasiperiodic structure in the nonlattice case.