Delone Sets: Local Rules, Countable Families and Periodicity

The concept of a Delone set of finite type, i.e., a Delone set in which only finite number of different clusters of a given size can occur, is a generally accepted geometric model of the atomic structure of solids, essentially of crystalline and quasicrystalline structures. The paper provides modified statements of local criteria in terms of local rules for crystalline structures, that is, for Delone sets with crystallographic symmetry group. The main result of the paper is a theorem stating that if a family of sets determined by local rules is finite or countable, then among these sets there is necessarily a crystal. From this theorem, in particular, it follows that if the local rules generate a single Delone set, then this set is a crystal. On the other hand, this theorem is equivalent to the statement that if in a family determined by the local rules there are no crystals, then it is uncountable. This statement implies, in particular, that there are uncountably many Penrose patterns.