Delone Sets: Local Identity and Global Symmetry

TL;DR

This paper proves a local criterion for crystalline structures in Delone sets, showing that locally antipodal sets exhibit global symmetry and crystallinity, extending understanding of local-global relations in geometric structures.

Contribution

It generalizes the local criterion for regular systems to include locally antipodal Delone sets, establishing their crystalline nature and conditions for regularity.

Findings

Locally antipodal Delone sets are crystalline.

If all 2R-clusters are identical, the set is a regular system.

The paper extends local criteria to broader classes of Delone sets.

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 crystallgraphic group. So-called locally antipodal Delone sets, i.e. such sets in which all 2R-clusters are centrally symmetrical, are considered. It turns out that the local antipodal sets have crystalline structure. Moreover, if in a locally antipodal set all 2R-clusters are the same the set is a regular system, i.e. a Delone set whose symmetry group operates transitively on the set.