Crystallographers use x-ray spectra to understand the structure of solids and ergodic theorists use spectra of abstract dynamical systems for structural studies; the relation of the two is shown here.

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We discuss a new general phenomenon pertaining to tiling models of quasicrystal growth. It is known that with Penrose tiles no (deterministic) local matching rules exist which guarantee defect-free tiling for regions of arbitrary large size. We prove that this property holds quite generally: namely, that the emergence of defects in quasicrystal growth is unavoidable for all aperiodic tiling models in the plane with local matching rules, and for many models in R 3 satisfying certain conditions.

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