Skip to main content
SpringerLink
Log in

Delone Sets and Tilings: Local Approach

  • Research Articles
  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

We present new results in the local theory of Delone sets, regular systems, and isogonal tilings. In particular, we prove a local criterion for isogonal tilings of the Euclidean space. This criterion is then applied to the study of \(2R\)-isometric Delone sets, where \(R\) is the covering radius for these sets. For regular systems in the plane we establish the exact value \(\widehat{\rho}_2=4R\) of the regularity radius. We prove that in any cell of the Delone tiling in an arbitrary Delone set in the plane, there is a vertex at which the local group is crystallographic. Hence, the subset of points with local crystallographic groups in a Delone set in the plane is itself a Delone set with covering radius at most \(2R\).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.
Fig. 15.
Fig. 16.
Fig. 17.
Fig. 18.
Fig. 19.
Fig. 20.
Fig. 21.
Fig. 22.
Fig. 23.
Fig. 24.
Fig. 25.
Fig. 26.
Fig. 27.
Fig. 28.

Similar content being viewed by others

Notes

  1. Initially, we proved the impossibility of isogonal tilings for each subcase separately. The general proof given here was proposed by K. O. Besov.

  2. Another way is to consider the \(m\)- and \((m+2)\)-stars centered at the point \(x\in X\) for sufficiently large \(m\in{\mathbb N}\). Then all the further arguments remain valid with \(N=3m(m+1)+1\).

References

  1. I. A. Baburin, M. Bouniaev, N. Dolbilin, N. Yu. Erokhovets, A. Garber, S. V. Krivovichev, and E. Schulte, “On the origin of crystallinity: A lower bound for the regularity radius of Delone sets,” Acta Crystallogr., Sect. A 74 (6), 616–629 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Delaunay, “Sur la sphère vide: A la mémoire de Georges Voronoï,” Izv. Akad. Nauk SSSR, Otd. Mat. Estestv. Nauk, No. 6, 793–800 (1934).

    MATH  Google Scholar 

  3. B. N. Delone, “Geometry of positive quadratic forms,” Usp. Mat. Nauk, No. 3, 16–62 (1937).

    Google Scholar 

  4. B. N. Delone, N. P. Dolbilin, and M. I. Shtogrin, “Combinatorial and metric theory of planigons,” Proc. Steklov Inst. Math. 148, 111–141 (1980) [transl. from Tr. Mat. Inst. Steklova 148, 109–140 (1978)].

    MATH  Google Scholar 

  5. B. N. Delone, N. P. Dolbilin, M. I. Shtogrin, and R. V. Galiulin, “A local criterion for regularity of a system of points,” Sov. Math., Dokl. 17 (2), 319–322 (1976) [transl. from Dokl. Akad. Nauk SSSR 227 (1), 19–21 (1976)].

    MATH  Google Scholar 

  6. N. P. Dolbilin, “Crystal criterion and locally antipodal Delaunay sets,” Vestn. Chelyab. Gos. Univ., No. 3, 6–17 (2015).

    MathSciNet  Google Scholar 

  7. N. P. Dolbilin, “Delone sets in \(\mathbb R^3\) with \(2R\)-regularity conditions,” Proc. Steklov Inst. Math. 302, 161–185 (2018) [transl. from Tr. Mat. Inst. Steklova 302, 176–201 (2018)].

    Article  MathSciNet  MATH  Google Scholar 

  8. N. P. Dolbilin, “From local identity to global order,” in Discrete Mathematics and Its Applications: Mater. XIII Lupanov Int. Semin. (Moscow State Univ., Moscow, 2019), pp. 13–22 [in Russian].

    Google Scholar 

  9. N. P. Dolbilin, “Local groups in Delone sets,” in Numerical Geometry, Grid Generation and Scientific Computing (Springer, Cham, 2021), Lect. Notes Comput. Sci. Eng. 143, pp. 3–11.

    Chapter  Google Scholar 

  10. N. Dolbilin, A. Garber, U. Leopold, E. Schulte, and M. Senechal, “On the regularity radius of Delone sets in \(\mathbb R^3\),” Discrete Comput. Geom. 66, 996–1024 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  11. N. P. Dolbilin and A. N. Magazinov, “Uniqueness theorem for locally antipodal Delaunay sets,” Proc. Steklov Inst. Math. 294, 215–221 (2016) [transl. from Tr. Mat. Inst. Steklova 294, 230–236 (2016)].

    Article  MathSciNet  MATH  Google Scholar 

  12. N. Dolbilin and D. Schattschneider, “The local theorem for tilings,” in Quasicrystals and Discrete Geometry: Proc. 1995 Fall Programme Fields Inst., Toronto, 1995 (Am. Math. Soc., Providence, RI, 1998), Fields Inst. Monogr. 10, pp. 193–199.

    Google Scholar 

  13. N. P. Dolbilin and M. I. Shtogrin, “Local groups in Delone sets: A conjecture and results,” Russ. Math. Surv. 76 (6), 1137–1139 (2021) [transl. from Usp. Mat. Nauk 76 (6), 193–194 (2021)].

    Article  MathSciNet  MATH  Google Scholar 

  14. N. P. Dolbilin and M. I. Shtogrin, “Crystallographic properties of local groups of a Delone set in a Euclidean plane,” Comput. Math. Math. Phys. 62 (8), 1265–1274 (2022) [transl. from Zh. Vychisl. Mat. Mat. Fiz. 62 (8), 1289–1299 (2022)].

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Hilbert and S. Cohn-Vossen, Anschauliche Geometrie (J. Springer, Berlin, 1932).

    Book  MATH  Google Scholar 

  16. F. Laves, “Ebenenteilung und Koordinationszahl,” Z. Kristallogr. 78, 208–241 (1931).

    Article  MATH  Google Scholar 

  17. M. I. Shtogrin, “Bound on the order of spider’s axis in a locally regular Delone system,” in Geometry, Topology, Algebra and Number Theory, Applications: Int. Conf. dedicated to the 120th anniversary of Boris Delone, Moscow, 2010: Abstracts (Steklov Math. Inst., Moscow, 2010), pp. 168–169 [in Russian].

    Google Scholar 

  18. A. Shubnikov, “On the structure of crystals. I,” Izv. Imp. Akad. Nauk 10 (9), 755–779 (1916).

    Google Scholar 

Download references

Acknowledgments

We are grateful to Konstantin Besov for a useful suggestion that allowed us to shorten the proof of Theorem 5.1. We are also grateful to the referees, whose comments helped to improve the manuscript, as well as to Aleksei Malkov for his help in preparing the figures.

Funding

The work of the first author (Section 2 and Subsections 1–3 and 5 in Section 5) was supported by the Russian Science Foundation under grant no. 20-11-19998, https://rscf.ru/project/20-11-19998/.

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    N. P. Dolbilin & M. I. Shtogrin

Authors
  1. N. P. Dolbilin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. I. Shtogrin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to N. P. Dolbilin.

Additional information

Translated from Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 318, pp. 73–98 https://doi.org/10.4213/tm4275.

Translated by I. Nikitin

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dolbilin, N.P., Shtogrin, M.I. Delone Sets and Tilings: Local Approach. Proc. Steklov Inst. Math. 318, 65–89 (2022). https://doi.org/10.1134/S0081543822040071

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543822040071

Search

Navigation