Delone Sets: Local Identity and Global Symmetry

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


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.


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



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.


  1. 1.
    B.N. Delone, N.P. Dolbilin, M.I. Stogrin, R.V. Galiuilin, A local criterion for regularity of a system of points. Soviet Math. Dokl. 17, 319–322 (1976)Google Scholar
  2. 2.
    E.S. Fedorov, Elements of the study of figures. Zap. Mineral. Imper. S.Peterburgskogo Obschestva 21(2), 1–279 (1985)Google Scholar
  3. 3.
    A. Schönflies, Kristallsysteme und Kristallstruktur (Druck und verlag von BG Teubner, Leipzig, 1891)Google Scholar
  4. 4.
    L. Bieberbach, Über die Bewegungsgruppen des n-dimensionalen Euklidischen Räumes I. Math. Ann. 70, 207–336; II. Math. Ann. 72(1912), 400–412 (1911)MathSciNetCrossRefGoogle Scholar
  5. 5.
    R. Feynman, R. Leighton, M. Sands, Feynman Lectures on Physics, vol. II (Addison-Wesley, Reading, MA, 1964)zbMATHGoogle Scholar
  6. 6.
    N.P. Dolbilin, J.C. Lagarias, M. Senechal, Multiregular point systems. Discr. Comput. Geom. 20, 477–498 (1998)MathSciNetCrossRefGoogle Scholar
  7. 7.
    N. Dolbilin, E. Schulte, The local theorem for monotypic tilings. Electron. J. Combin. 11, 2 (2004). (Research Paper 7, 19pp)MathSciNetzbMATHGoogle Scholar
  8. 8.
    N. Dolbilin, E. Schulte, A local characterization of combinatorial multihedrality in tilings. Contrib. Discrete Math. 4(1), 1–11 (2009)MathSciNetzbMATHGoogle Scholar
  9. 9.
    B. Delaunay, Sur la sphère vide. A la mémoire de Georges Voronoï. Bull. de l’Académie des Sci. de l’URSS 6, 793–800 (1934)zbMATHGoogle Scholar
  10. 10.
    B.N. Delone, Geometry of positive quadratic forms, Uspekhi Matem. Nauk 3, 16–62 (1937). (in Russian)Google Scholar
  11. 11.
    N. Dolbilin, delone sets in \(\mathbb{R}^3\): regularity conditions. Fundam. Appl. Math. 21, (6) (2016) (in Russian, English translation will appear in Journal of Mathematical Sciences in 2018)Google Scholar
  12. 12.
    D. Schattschneider, N. Dolbilin, One corona is enough for the Euclidean plane, in Quasicrystals and Discrete Geometry, Fields Inst. Monogr., vol 10 (American Mathematical Society, Providence RI, 1998), pp. 207–246Google Scholar
  13. 13.
    N.P. Dolbilin, A criterion for crystal and locally antipodal Delone sets. Vestnik Chelyabinskogo Gos. Universiteta 3(358), 6–17 (2015). (in Russian)Google Scholar
  14. 14.
    N.P. Dolbilin, A.N. Magazinov, Locally antipodal Delauney sets. Russian Math. Surveys. 70(5), 958–960 (2015)MathSciNetCrossRefGoogle Scholar
  15. 15.
    N.P. Dolbilin, M.I. Shtogrin, A local criterion for a crystal structure, in Abstracts of the IXth All-Union Geometrical Conference (Kishinev, 1988) p. 99 (in Russian)Google Scholar

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© 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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