Deceptions in quasicrystal growth

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 inR ³ satisfying certain conditions.Research supported in part by NSF Grant No. DMS-9304269 and Texas ARP Grants 003658113 and 003658007.