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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