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

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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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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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Authors and Affiliations

  1. Aix Marseille Université, Université de Toulon, CNRS, CPT UMR 7332, 13288, Marseille, France

    Oleg Ogievetsky & Senya Shlosman

  2. Institute of the Information Transmission Problems, RAS, Moscow, Russia

    Senya Shlosman

  3. Skolkovo Institute of Science and Technology, Moscow, Russia

    Senya Shlosman

  4. Kazan Federal University, Kremlevskaya 17, Kazan, Russia, 420008

    Oleg Ogievetsky

  5. Lebedev Physical Institute of the Russian Academy of Sciences, 53, Leninskiy Prospekt, Moscow, Russia, 119991

    Oleg Ogievetsky

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  1. Oleg Ogievetsky
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  2. Senya Shlosman
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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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  • DOI: https://doi.org/10.1007/s00454-019-00064-3

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