The Six Cylinders Problem: \(\mathbb {D}_{3}\)-Symmetry Approach

Abstract

Motivated by a question of W. Kuperberg, we study the 18-dimensional manifold of configurations of six non-intersecting infinite cylinders of radius r, all touching the unit ball in \(\mathbb {R}^{3}\). We find a configuration with

$$\begin{aligned} r=\frac{1}{8}\Big ( 3+\sqrt{33}\Big ) \approx 1.093070331. \end{aligned}$$

We believe that this value is the maximum possible.

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References

  1. 1.

    Braß, P., Wenk, C.: On the number of cylinders touching a ball. Geom. Dedic. 81(1–3), 281–284 (2000)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Conway, J.H., Radin, C., Sadun, L.: On angles whose squared trigonometric functions are rational. Discret. Comput. Geom. 22(3), 321–332 (1999)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Coxeter, H.S.M.: Regular Polytopes, 3rd edn. Dover, New York (1973)

    Google Scholar 

  4. 4.

    Fejes Tóth, L.: Lagerungen in der Ebene auf der Kugel und im Raum. Die Grundlehren der mathematischen Wissenschaften, 2nd edn. Springer, Berlin (1972)

    Google Scholar 

  5. 5.

    Firsching, M.: Optimization Methods in Discrete Geometry. PhD thesis, Freie Universität Berlin, Berlin (2016)

  6. 6.

    Heppes, A., Szabó, L.: On the number of cylinders touching a ball. Geom. Dedic. 40(1), 111–116 (1991)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kuperberg, W.: How many unit cylinders can touch a unit ball? Problem 3.3. In: DIMACS Workshop on Polytopes and Convex Sets. Rutgers University (1990)

  8. 8.

    Kusner, R., Kusner, W., Lagarias, J.C., Shlosman, S.: Configuration spaces of equal spheres touching a given sphere: the twelve spheres problem. In: Ambrus, G., Bárány, I., Böröczky, K.J., Fejes Tóth, G., Pach, J. (eds.) New Trends in Intuitive Geometry. Bolyai Society Mathematical Studies, vol 27, pp. 219–277. Springer, Berlin (2018) (arXiv:1611.10297)

  9. 9.

    Ogievetsky, O., Shlosman, S.: Extremal cylinder configurations I: configuration \(C_{\mathfrak{m}}\) (2018). arXiv:1812.09543 [math.MG]

  10. 10.

    Ogievetsky, O., Shlosman, S.: Extremal cylinder configurations II: configuration \(O_6\) (to appear)

  11. 11.

    Schütte, K., van der Waerden, B.L.: Das problem der dreizehn Kugeln. Math. Ann. 125, 325–334 (1953)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Wolfram Research, Inc., Mathematica, Version 11.3, Champaign, IL (2018)

Download references

Acknowledgements

Part of the work of S. S. has been carried out in the framework of the Labex Archimede (ANR-11-LABX-0033) and of the A*MIDEX Project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French Government programme managed by the French National Research Agency (ANR). Part of the work of S. S. has been carried out at IITP RAS. The support of Russian Foundation for Sciences (Project No. 14-50-00150) is gratefully acknowledged by S. S. The work of O. O. was supported by the Program of Competitive Growth of Kazan Federal University and by the Grant RFBR 17-01-00585.

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Correspondence to Senya Shlosman.

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Ogievetsky, O., Shlosman, S. The Six Cylinders Problem: \(\mathbb {D}_{3}\)-Symmetry Approach. Discrete Comput Geom 65, 385–404 (2021). https://doi.org/10.1007/s00454-019-00064-3

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Keywords

  • Critical configuration
  • Unlocking procedure
  • Integrability

Mathematics Subject Classification

  • 05B40
  • 52C25