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


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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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).

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  • Critical configuration
  • Unlocking procedure
  • Integrability

Mathematics Subject Classification

  • 05B40
  • 52C25