Structural Chemistry

, Volume 27, Issue 6, pp 1725–1732 | Cite as

Delone sets with congruent clusters

Original Research

Abstract

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.

Keywords

Delone sets Regular systems Multi-regular system Crystallographic groups Clusters Local approach 

Notes

Acknowledgments

The author is grateful to Dr. Igor Baburin for his helpful suggestion to consider Delone sets with centrally symmetrical patches and to Dr. Andrei Ordine for his help in editing the English text. This work is supported by the Russian Science Foundation under Grant 14-11-00414.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Steklov Mathematical Institute of the Russian Academy of SciencesMoscowRussia

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