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

Oleg Ogievetsky (1, 2) , Senya Shlosman (1, 2)

1 CPT - Centre de Physique Théorique - UMR 7332
2 CPT - E2 Géométrie, Physique et Symétries

We study the path $\Gamma=\{ C_{6,x}\ \vert\ x\in [0,1]\}$ in the moduli space of configurations of 6 equal cylinders touching the unit sphere. Among the configurations $C_{6,x}$ is the record configuration $C_{\mathfrak{m}}$ of \cite{OS}. 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 $\frac{(1 + x) (1 + 3 x)}{3}$ 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.

Domaines: Géométrie métrique [math.MG]

Oleg Ogievetsky, Senya Shlosman. Extremal Cylinder Configurations I: Configuration $C_{\mathfrak{m}}$. Discrete and Computational Geometry, 2021, 66 (1), pp.140-164. ⟨10.1007/s00454-020-00244-6⟩. ⟨hal-02003983⟩