Local rules for quasicrystals

The relationship of local ordering and long-range order is studied for quasicrystalline tilings of plane and space. Two versions of the concept of local rules are introduced: strong and weak. Necessary conditions of the existence of strong local rules are found. They are mainly reduced to the constraints for irrational numbers related to incommensurabilities of the quasicrystals. For planar quasicrystals the quadratic irrationalities play an important role. For three-dimensional quasicrystals not only quadratic but also cubic irrationalities are allowed. The existence of weak local rules is established for almost all two-dimensional quasicrystals based on quadratic irrationalities and for the three-dimensional quasicrystal having icosahedral symmetry.