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

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.