Extremal Cylinder Configurations I: Configuration $C_{\mathfrak {m}}$

I do not ask for better than not to be believed. Axel Munthe, The story of San Michele

Abstract

We study the path $\Gamma =\{ C_{6,x}\mid x\in [0,1]\}$ in the moduli space of configurations of six equal cylinders touching the unit sphere. Among the configurations $C_{6,x}$ is the record configuration $C_{\mathfrak {m}}$ of Ogievetsky and Shlosman (Discrete Comput Geom 2019, https://doi.org/10.1007/s00454-019-00064-3). We show that $C_{\mathfrak {m}}$ is a local sharp maximum of the distance function, so in particular the configuration $C_{\mathfrak {m}}$ is not only unlockable but rigid. We show that if ${(1 + x) (1 + 3 x)}/{3}$ is a rational number but not a square of a rational number, the configuration $C_{6,x}$ has some hidden symmetries, part of which we explain.