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

## TL;DR

This paper analyzes a specific family of six-cylinder configurations touching a sphere, demonstrating that the record configuration is a local maximum and exploring hidden symmetries in certain cases.

## Contribution

It proves that the record configuration is a rigid local maximum and identifies conditions under which configurations exhibit hidden symmetries.

## Key findings

- Record configuration $C_{\mathfrak{m}}$ is a local sharp maximum.
- Configuration $C_{\mathfrak{m}}$ is rigid and unlockable.
- Configurations with certain rational parameters have hidden symmetries.

## Abstract

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.