Mutually touching infinite cylinders in the 3D world of lines

TL;DR

This paper investigates the maximum number of mutually touching infinite cylinders in 3D, establishing an upper bound of 10 for smooth cross-section cylinders and exploring configurations of elliptic cylinders.

Contribution

It introduces topological classification tools and provides new bounds on the number of mutually touching cylinders, including numerical configurations and properties of the chirality matrix.

Findings

Maximum of 10 mutually touching cylinders with smooth cross-sections.

Numerical configurations of 8 and 9 elliptic cylinders.

Properties of the chirality matrix related to graph theory.

Abstract

Recently we gave arguments that only two unique topologically different configurations of 7 equal all mutually touching round cylinders (the configurations being mirror reflections of each other) are possible in 3D, although a whole world of configurations is possible already for round cylinders of arbitrary radii. It was found that as many as 9 round cylinders (all mutually touching) are possible in 3D while the upper bound for arbitrary cylinders was estimated to be not more than 14 under plausible arguments. Now by using the chirality and Ring matrices that we introduced earlier for the topological classification of line configurations, we have given arguments that the maximal number of mutually touching straight infinite cylinders of arbitrary cross-section (provided that its boundary is a smooth curve) in 3D cannot exceed 10. We generated numerically several configurations of 10 cylinders, restricting ourselves with elliptic cylinders. Configurations of 8 and 9 equal elliptic cylinders (all in mutually touching) are generated numerically as well. A possibility and restriction of continuous transformations from elliptic into round cylinder configurations are discussed. Some curious results concerning the properties of the chirality matrix (which coincides with Seidel's adjacency matrix important for the Graph theory) are presented.