When Periodicities Enforce Aperiodicity

Non-periodic tilings and local rules are commonly used to model the long range aperiodic order of quasicrystals and the finite-range energetic interactions that stabilize them. This paper focuses on planar rhombus tilings, which are tilings of the Euclidean plane, which can be seen as an approximation of a real plane embedded in a higher dimensional space. Our main result is a characterization of the existence of local rules for such tilings when the embedding space is four-dimensional. The proof is an interplay of algebra and geometry that makes use of the rational dependencies between the coordinates of the embedded plane. We also apply this result to some cases in a higher dimensional embedding space, notably tilings with n-fold rotational symmetry.