Delone Sets: Local Identity and Global Symmetry

Abstract

In the paper we present a proof of the local criterion for crystalline structures which generalizes the local criterion for regular systems. A Delone set is called a crystal if it is invariant with respect to a crystallographic group. Locally antipodal Delone sets, i.e. those in which all 2R-clusters are centrally symmetrical, are considered and we prove that they have crystalline structure. Moreover, if in a locally antipodal set all 2R-clusters are the same, then the set is a regular system, i.e. a Delone set whose symmetry group operates transitively on the set.