Symmetry, topology and the maximum number of mutually pairwise touching infinite cylinders: complete configuration classification

TL;DR

This paper classifies all possible arrangements of infinite cylinders in 3D space where each touches every other, establishing a maximum number and explaining the unique configuration of seven equal cylinders, with implications for graph theory.

Contribution

It provides a complete classification of mutually touching infinite cylinders in 3D and explains the uniqueness of the seven-cylinder configuration, advancing geometric and graph theoretical understanding.

Findings

Maximum number of mutually touching cylinders in 3D is established.

Unique configuration of seven equal cylinders is explained.

Results relate to the theory of graphs via the chirality matrix.

Abstract

We provide a complete classification of possible configurations of mutually pairwise touching infinite cylinders in Euclidian 3D space. It turns out that there is a maximum number of such cylinders possible in 3D independently on the shape of the cylinder cross-sections. We give the explanation of the uniqueness of the non-trivial configuration of seven equal mutually touching round infinite cylinders found earlier. Some results obtained for the chirality matrix which is equivalent to the Seidel adjacency matrix may be found useful for the theory of graphs.