Skip to main content
SpringerLink
Log in
Book cover

New Trends in Intuitive Geometry pp 25–47Cite as

Contact Numbers for Sphere Packings

  • Chapter
  • First Online:

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 27))

Abstract

In discrete geometry, the contact number of a given finite number of non-overlapping spheres was introduced as a generalization of Newton’s kissing number. This notion has not only led to interesting mathematics, but has also found applications in the science of self-assembling materials, such as colloidal matter. With geometers, chemists, physicists and materials scientists researching the topic, there is a need to inform on the state of the art of the contact number problem. In this paper, we investigate the problem in general and emphasize important special cases including contact numbers of minimally rigid and totally separable sphere packings. We also discuss the complexity of recognizing contact graphs in a fixed dimension. Moreover, we list some conjectures and open problems.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   99.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD   129.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Notes

  1. 1.

    The complete list (up to possible omissions due to round off errors) of minimally rigid packings of \(n\le 9\) spheres and a preliminary list of \(n=10\) spheres appears on the arXiv [3]. The paper [3] only contains a partial list, so for the more complete list we refer to the arXiv version.

  2. 2.

    According to [3], for \(n\le 7\), it is possible to solve the system (3) using standard algebraic geometry methods for all \(A\in \mathcal {A}\) without filtering by geometric rules. Arkus et al. [3] attempted this using the package SINGULAR. Therefore, most likely for \(n=6,7\), the maximal contact graphs as obtained in [3] are optimal for minimally rigid sphere packings.

  3. 3.

    It is worth-noting that Koebe’s paper was written in German and titled ‘Kontaktprobleme der konformen Abbildung’ (Contact problems of conformal mapping). Andreev’s paper appeared in Russian. Probably, the first instance of this result appearing in English was in Thurston’s lecture notes that were distributed by the Princeton University in 1980. However, the lectures were delivered in 1978–1979 [42].

References

  1. L. Alonso, R. Cerf, The three dimensional polyominoes of minimal area. Electr. J. Combin. 3 (1996). #R27

    Google Scholar 

  2. E.M. Andreev, Convex polyhedra of finite volume in Lobačevskiĭ space. Mat. Sb. (N.S.) 83(125), 256–260 (1970). (Russian)

    MathSciNet  Google Scholar 

  3. N. Arkus, V.N. Manoharan, M.P. Brenner, Deriving finite sphere packings. SIAM J. Discret. Math. 25(4), 1860–1901 (2011), arXiv:1011.5412v2 [cond-mat.soft]

    Article  MathSciNet  Google Scholar 

  4. K. Ball, An elementary introduction to modern convex geometry, in Flavors of Geometry, vol. 31, Mathematical Sciences Research Institute Publications, ed. by S. Levy (Cambridge University Press, Cambridge, 1997), pp. 1–58

    Google Scholar 

  5. U. Betke, M. Henk, J.M. Wills, Finite and infinite packings. J. reine angew. Math. 53, 165–191 (1994)

    MathSciNet  MATH  Google Scholar 

  6. A. Bezdek, Locally separable circle packings. Studia Sci. Math. Hungar. 18(2–4), 371–375 (1983)

    MathSciNet  MATH  Google Scholar 

  7. K. Bezdek, On the maximum number of touching pairs in a finite packing of translates of a convex body. J. Combin. Theory Ser. A 98, 192–200 (2002)

    Article  MathSciNet  Google Scholar 

  8. K. Bezdek, Contact numbers for congruent sphere packings in Euclidean 3-space. Discret. Comput. Geom. 48(2), 298–309 (2012)

    Article  MathSciNet  Google Scholar 

  9. K. Bezdek, Lectures on Sphere Arrangements - the Discrete Geometric Side, vol. 32, Fields Institute Monographs (Springer, New York, 2013)

    Book  Google Scholar 

  10. K. Bezdek, P. Brass, On \(k^+\)-neighbour packings and one-sided Hadwiger configurations. Beitr. Algebr. Geom. 44, 493–498 (2003)

    MathSciNet  MATH  Google Scholar 

  11. K. Bezdek, S. Reid, Contact graphs of unit sphere packings revisited. J. Geom. 104(1), 57–83 (2013)

    Article  MathSciNet  Google Scholar 

  12. K. Bezdek, Zs. Lángi, Density bounds for outer parallel domains of unit ball packings. Proc. Steklov Inst. Math. 288/1, 209–225 (2015)

    Article  MathSciNet  Google Scholar 

  13. K. Bezdek, R. Connelly, G. Kertész, On the average number of neighbours in spherical packing of congruent circles, Intuitive Geometry, vol. 48, Colloquia Mathematica Societatis János Bolyai (North Holland, Amsterdam, 1987), pp. 37–52

    Google Scholar 

  14. K. Bezdek, B. Szalkai, I. Szalkai, On contact numbers of totally separable unit sphere packings. Discret. Math. 339(2), 668–676 (2015)

    Article  MathSciNet  Google Scholar 

  15. L. Bowen, Circle packing in the hyperbolic plane. Math. Phys. Electr. J. 6, 1–10 (2000)

    MathSciNet  MATH  Google Scholar 

  16. P. Boyvalenkov, S. Dodunekov, O. Musin, A survey on the kissing numbers. Serdica Math. J. 38(4), 507–522 (2012)

    MathSciNet  MATH  Google Scholar 

  17. P. Brass, Erdős distance problems in normed spaces. Comput. Geom. 6, 195–214 (1996)

    Article  MathSciNet  Google Scholar 

  18. H. Breu, D.G. Kirkpatrick, On the complexity of recognizing intersection and touching graphs of discs, in Graph Drawing ed. by F.J. Brandenburg, Proceedings of Graph Drawing 95, Passau, September 1995. Lecture Notes in Computer Science, vol. 1027, Springer, Berlin, (1996), pp. 88–98

    Chapter  Google Scholar 

  19. P. Erdős, On sets of distances of \(n\) points. Am. Math. Mon. 53, 248–250 (1946)

    Google Scholar 

  20. P. Erdős, Problems and results in combinatorial geometry, in Discrete Geometry and Convexity, vol. 440, Annals of the New York Academy of Sciences, ed. by J.E. Goodman, et al. (vvv, bbb, 1985), pp. 1–11

    Article  MathSciNet  Google Scholar 

  21. G. Fejes Tóth, L. Fejes Tóth, On totally separable domains. Acta Math. Acad. Sci. Hungar. 24, 229–232 (1973)

    Article  MathSciNet  Google Scholar 

  22. H. Hadwiger, Über Treffenzahlen bei translations gleichen Eikörpern. Arch. Math. 8, 212–213 (1957)

    Article  MathSciNet  Google Scholar 

  23. T.C. Hales, A proof of the Kepler conjecture. Ann. Math. 162(2–3), 1065–1185 (2005)

    Article  MathSciNet  Google Scholar 

  24. F. Harary, H. Harborth, Extremal animals. J. Comb. Inf. Syst. Sci. 1(1), 1–8 (1976)

    MathSciNet  MATH  Google Scholar 

  25. H. Harborth, Lösung zu problem 664A. Elem. Math. 29, 14–15 (1974)

    Google Scholar 

  26. H. Harborth, L. Szabó, Z. Ujvári-Menyhárt, Regular sphere packings. Arch. Math. (Basel) 78/1, 81–89 (2002)

    Article  MathSciNet  Google Scholar 

  27. B. Hayes, The science of sticky spheres. Am. Sci. 100, 442–449 (2012)

    Article  Google Scholar 

  28. B. Hayes, Sphere packings and hamiltonian paths (blog post posted on 13 March 2013), http://bit-player.org/2013/sphere-packings-and-hamiltonian-paths

  29. P. Hliněný, Touching graphs of unit balls, in Graph Drawing ed. by G. DiBattista, Proceedings of Graph Drawing 97, Rome, September. Lecture Notes in Computer Science, vol. 1353 (Springer, Berlin, 1997), pp. 350–358

    Chapter  Google Scholar 

  30. P. Hliněný, J. Kratochvíl, Representing graphs by disks and balls (a survey of recognition-complexity results). Discret. Math. 229, 101–124 (2001)

    Article  MathSciNet  Google Scholar 

  31. M. Holmes-Cerfon, Enumerating nonlinearly rigid sphere packings. SIAM Rev. 58(2), 229–244 (2016), arXiv:1407.3285v2 [cond-mat.soft]

    Article  MathSciNet  Google Scholar 

  32. R.S. Hoy, J. Harwayne-Gidansky, C.S. O’Hern, Structure of finite sphere packings via exact enumeration: implications for colloidal crystal nucleation. Phys. Rev. E 85, 051403 (2012)

    Article  Google Scholar 

  33. G.A. Kabatiansky, V.I. Levenshtein, Bounds for packings on a sphere and in space. Probl. Pereda. Inf. 14, 3–25 (1978)

    MathSciNet  Google Scholar 

  34. G. Kertész, On totally separable packings of equal balls. Acta Math. Hungar. 51(3-4), 363–364 (1988)

    Article  MathSciNet  Google Scholar 

  35. P. Koebe, Kontaktprobleme der konformen Abbildung. Ber. Verh. Sächs. Akad. Leipzig 88, 141–164 (1936). (German)

    MATH  Google Scholar 

  36. G. Kuberberg, O. Schramm, Average kissing numbers for non-congruent sphere packings. Math. Res. Lett. 1, 339–344 (1994)

    Article  MathSciNet  Google Scholar 

  37. V.N. Manoharan, Colloidal matter: packing, geometry, and entropy. Science 349, 1253751 (2015)

    Article  MathSciNet  Google Scholar 

  38. J.C. Maxwell, On the calculation of the equilibrium and stiffness of frames. Philos. Mag. 27, 294–299 (1864)

    Article  Google Scholar 

  39. O.R. Musin, The kissing number in four dimensions. Ann. Math. (2) 168/1, 1–32 (2008)

    Article  MathSciNet  Google Scholar 

  40. A.M. Odlyzko, N.J.A. Sloane, New bounds on the number of unit spheres that can touch a unit sphere in \(n\)-dimensions. J. Comb. Theory, Ser. A 26, 210–214 (1979)

    Article  MathSciNet  Google Scholar 

  41. K. Schütte, B.L. Van Der Waerden, Das Problem der dreizehn Kugeln. Math. Ann. 125, 325–334 (1953)

    Article  MathSciNet  Google Scholar 

  42. W. Thurston, The geometry and topology of 3-manifolds, Princeton Lecture Notes (1980)

    Google Scholar 

Download references

Acknowledgements

The first author is partially supported by a Natural Sciences and Engineering Research Council of Canada Discovery Grant. The second author is supported by a Vanier Canada Graduate Scholarship (NSERC), an Izaak Walton Killam Memorial Scholarship and Alberta Innovates Technology Futures (AITF). The authors would like to thank the anonymous referee for careful reading and an interesting reference.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, Canada

    Károly Bezdek & Muhammad A. Khan

  2. Department of Mathematics, University of Pannonia, Veszprém, Hungary

    Károly Bezdek

Authors
  1. Károly Bezdek
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Muhammad A. Khan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Károly Bezdek .

Editor information

Editors and Affiliations

  1. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gergely Ambrus

  2. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Imre Bárány

  3. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Károly J. Böröczky

  4. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    Gábor Fejes Tóth

  5. MTA Alfréd Rényi Institute of Mathematics, Budapest, Hungary

    János Pach

Rights and permissions

Reprints and permissions

Copyright information

© 2018 János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Bezdek, K., Khan, M.A. (2018). Contact Numbers for Sphere Packings. In: Ambrus, G., Bárány, I., Böröczky, K., Fejes Tóth, G., Pach, J. (eds) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57413-3_2

Download citation

Publish with us

Policies and ethics

Search

Navigation