Local rules and growth in quasicrystals

(Received 5 April 1990, revised 26 June 1990, accepted 17 September 1990)

Abstract
The problem of local definition of general projected quasicrystals (QC) in two and three dimensions is studied. It is shown that local order in projected QC is a result of the existence of QC lines of points within it (which can be connected to the incommensurate constants of the QC and its symmetry) and of a special definition of the strips. The shape of the strip will determine whether strong, weak or no local rules will exist in such a QC. For the non crystallographic lattices with n-fold symmetry and for the icosahedral lattice we contruct lattices with strong local rules and prove that strong local rules indeed exist. We suggest that those results can serve to explain the growth process and the relative stability of quasicrystals.