Delone sets with congruent clusters

In the paper, we present a short survey and new results of the local theory of regular systems, more precisely the part that establishes for a given Delone set the link between congruence of fragments (‘clusters’) of the set and its global symmetry. The theory describes ‘local rules’ of a Delone set which imply its regularity or multi-regularity, i.e., imply that a Delone set is an orbit or the union of several orbits, respectively, under its symmetry group. We will discuss a cycle of new results of the local theory for Delone sets which have centrally symmetrical fragments: clusters or patches.