Crystal criterion and antipodal Delaunay sets

It is proved that a discrete set of points repeatability local configurations under certain conditions implies the so-called «global order», which includes the presence of a plurality of crystallographic symmetry group. It is also proved that the set of Delaunay, in which all $2R$-clusters are antipodal, that is centrally symmetric, is itself a centrally symmetric with respect to each of its points. Moreover, if in addition to this cluster are identical, then the set is correct, i. e. its symmetry group acts transitively.
This article based on a lecture delivered at the International Conference «Quantum topology» (5-17 July 2014), organized by the Laboratory of Quantum Topology of Chelyabinsk State University.