Extremal Cylinder Configurations I: Configuration $$C_{\mathfrak {m}}$$ C m
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Extremal Cylinder Configurations I: Configuration Cm Oleg Ogievetsky1,4 · Senya Shlosman1,2,3 Received: 31 January 2019 / Revised: 14 June 2020 / Accepted: 16 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the path = {C6,x | x ∈ [0, 1]} in the moduli space of configurations of six equal cylinders touching the unit sphere. Among the configurations C6,x is the record configuration Cm of Ogievetsky and Shlosman (Discrete Comput Geom 2019, https:// doi.org/10.1007/s00454-019-00064-3). We show that Cm is a local sharp maximum of the distance function, so in particular the configuration Cm is not only unlockable but rigid. We show that if (1 + x)(1 + 3x)/3 is a rational number but not a square of a rational number, the configuration C6,x has some hidden symmetries, part of which we explain.
I do not ask for better than not to be believed. Axel Munthe, The story of San Michele
Keywords Critical configuration · Unlocking procedure · Integrability Mathematics Subject Classification 05B40 · 52C25
Editor in Charge: Kenneth Clarkson Oleg Ogievetsky [email protected] Senya Shlosman [email protected] 1
Aix Marseille Université, Université de Toulon, CNRS, CPT UMR 7332, 13288 Marseille, France
2
Skolkovo Institute of Science and Technology, Moscow, Russia
3
Institute of the Information Transmission Problems, RAS, Moscow, Russia
4
I.E. Tamm Department of Theoretical Physics, P.N. Lebedev Physical Institute, Leninsky prospekt 53, Moscow, Russia 119991
123
Discrete & Computational Geometry
1 Introduction This is a continuation of our work [3]. In that paper we were considering the configurations of six (infinite) nonintersecting cylinders of the same radius r touching the unit sphere S2 ⊂ R3 . We were interested in the maximal value of r for which this is possible. We have constructed in [3] the ‘record’ configuration Cm of six cylinders of radius rm =
3+
√
33
8
≈ 1.093070331.
(1)
Thus we know that the maximal value of r is at least rm . We believe that rm is in fact the maximal possible value for r , but we have no proof of that. In [3] we have constructed the deformation C6,x of the configuration C6 of six vertical unit nonintersecting cylinders. The configuration C6 corresponds to x = 1 while Cm —to x = 1/2. These configurations are shown in Fig. 1 (the green unit ball is in the center). To explain the results of the present paper, we introduce some notation. A cylinder ς touching the unit sphere S2 has a unique generator (a line parallel to the axis of the cylinder) ι(ς ) touching S2 . We will usually represent a configuration {ς1 , . . . , ς L } of cylinders touching the unit sphere by the configuration {ι(ς1 ), . . . , ι(ς L )} of lines tangent to S2 . The manifold of all such six-tuples we denote by M 6 . For example, let C6 ≡ C6 (0, 0, 0) be the configuration of six nonintersecting cylinders of radius 1, parallel to the z direction in R3 and touching the unit ball centered at the origin. The configuration of tangent lines associated to the c
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