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Delone Sets: Local Identity and Global Symmetry

  • Nikolay Dolbilin
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 234)

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.

Keywords

Delone (Delaunay) set Regular system Crystal Locally antipodal set Crystallographic group Symmetry group Cluster Local criterion for crystals Cluster counting function 

Notes

Acknowledgements

The author thanks Nikolay Andreev (Moscow) for making drawings and Andrey Ordine (Toronto) for his help in editing the English text. The author is very grateful to the anonymous reviewer for having made numerous significant comments that helped to improve this paper.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

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

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