  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.
