Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Theory of Combinatorial Algorithms
Prof. Emo Welzl and Prof. Bernd Gärtner
Department of Computer Science | Institute of Theoretical Computer Science | CADMO
Prof. Emo Welzl and Prof. Bernd Gärtner
Discrete geometric models for quasicrystals
Jeffrey C. Lagarias, AT&T Labs
Quasicrystalline materials, first discovered in 1984, are materials whose atomic structure has long-range translational order exhibited by X-ray diffraction patterns having sharp spots, but these patterns exhibit symmetries forbidden to crystals, e.g. icosahedral symmetry. The atomic structure of such materials must be aperiodic. This talk describes discrete geometric models for aperiodic structures including quasicrystals. A Delone set or (r, R)-set in Euclidean n-space is a discrete set which has a positive finite packing radius r and covering radius R by equal spheres. A Delone set of finite type is a Delone set X such that its interpoint distance vector X - X is a discrete closed set. This class of point sets includes Meyer sets, cut-and-project sets and most proposed models for quasicrystalline structures, including random tiling models. Such sets can be described by an ``address function'' which involves a finite number of extra ``internal dimensions'' in a way resembling that used in defining cut-and-project sets. Such sets are characterized by ``local rules'' in the sense that they have only a finite number of local configurations of the fixed radius 2R, and conversely. We geometrically characterize certain subclasses of such sets.
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