The Six Cylinders Problem: $\mathbb{D}_{3}$-symmetry Approach

TL;DR

This paper investigates the configuration of six non-intersecting infinite cylinders touching a unit ball in three-dimensional space, identifying a specific radius and proposing its maximality based on symmetry considerations.

Contribution

The authors introduce a novel symmetry-based approach to analyze the six cylinders problem and determine a specific configuration with a conjectured maximal radius.

Findings

Identified a configuration with radius approximately 1.09307.

Proposed that this radius is the maximal possible for such configurations.

Developed a $ ext{D}_3$-symmetry framework for the problem.

Abstract

Motivated by a question of W. Kuperberg, we study the 18-dimensional manifold of configurations of 6 non-intersecting infinite cylinders of radius $r,$ all touching the unit ball in $\mathbb{R}^{3}.$ We find a configuration with \[ r=\frac{1}{8}\left( 3+\sqrt{33}\right) \approx1.093070331\ .\] We believe that this value is the maximal possible.