Skip to main content
SpringerLink
Log in

Multiple lattice packings and coverings of spheres

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

Let Λ be a lattice inR n. Consider the systemS of unit spheres centered at the lattice points of Λ.S is called ak-fold lattice packing (covering) if each point inR n lies in at most (least)k of the open (closed) spheres ofS. Letd n k (D n k ) be the density of the closest (thinnest)k-fold lattice packing (covering) ofR n. After dealing several cases left by G. Fejes Tóth and A. Florian, we have concluded thatd n k >kd n 1 for all (n, k) (n≥2,k≥2) except (2, 2), (2, 3), (2, 4); andD 3 k <k D 31 for allk≥2.

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

Similar content being viewed by others

References

  1. Bambah, R. P.: On lattice coverings by spheres. Proc. Nat. Inst. Sci. India20, 25–52 (1954).

    Google Scholar 

  2. Blundon, W. J.: Multiple packing of cycles in the plane. J. London Math. Soc.38, 176–182 (1963).

    Google Scholar 

  3. Cassels, J. W. S.: An Introduction to the Geometry of Numbers. Berlin-Heidelberg-New York: Springer. 1971.

    Google Scholar 

  4. Danzer, L.: Drei Beispiele zu Lagerungsproblemen. Arch. Math.11, 159–165 (1960).

    Google Scholar 

  5. Fejes Tóth, G., andA. Florian: Mehrfache gitterförmige Kreis- und Kugelanordnungen. Mh. Math.79, 13–20 (1975).

    Google Scholar 

  6. Few, L.: Multiple packing of spheres: A survey. Proc. Coll. Convexity, Copenhagen 1965, 88–93. Copenhagen: Kobenhavns Univ. Math. Inst. 1967.

    Google Scholar 

  7. Few, L.: Double covering with spheres. Mathematika14, 207–214 (1967).

    Google Scholar 

  8. Few, L., andP. Kanagasabapathy: The double packing of spheres. J. London Math. Soc.44, 141–146 (1969).

    Google Scholar 

  9. Heppes, A.: Mehrfache gitterförmige Kreislagerungen in der Ebene. Acta Math. Acad. Sci. Hungar.10, 141–148 (1959).

    Google Scholar 

  10. Korkine, A., andG. Zolotareff: Sur les formes quadratique positive quaternaires. Math. Ann.5, 581–583 (1872).

    Google Scholar 

  11. Korkine, A., andG. Zolotareff: Sur les formes quadratiques positives. Math. Ann.11, 242–292 (1877).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Ohio State University at Marion, 43302, Marion, OH, U.S.A.

    L. J. Yang

Authors
  1. L. J. Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, L.J. Multiple lattice packings and coverings of spheres. Monatshefte für Mathematik 89, 69–76 (1980). https://doi.org/10.1007/BF01571566

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01571566

Keywords

Search

Navigation