Dense Packings with Nonparallel Cylinders

TL;DR

This paper investigates the maximum density of nonparallel cylinder packings in 3D space, demonstrating that such packings can approach the density of parallel packings locally and can achieve a global density of 1/2, surpassing previous bounds.

Contribution

It proves the existence of nonparallel cylinder packings with densities close to parallel packings and constructs a packing with a global density of 1/2, improving prior results.

Findings

Nonparallel packings can approach the density of parallel packings locally.

A nonparallel packing with a global density of 1/2 exists.

This work surpasses previous density bounds for nonparallel cylinder packings.

Abstract

A \emph{cylinder packing} is a family of congruent infinite circular cylinders with mutually disjoint interiors in $3$-dimensional Euclidean space. The \emph{local density} of a cylinder packing is the ratio between the volume occupied by the cylinders within a given sphere and the volume of the entire sphere. The \emph{global density} of the cylinder packing is obtained by letting the radius of the sphere approach infinity. It is known that the greatest global density is obtained when all cylinders are parallel to each other and each cylinder is surrounded by exactly six others. In this case, the global density of the cylinder packing equals $\pi/\sqrt{12}= 0.90689\ldots$. The question is how large a density can a cylinder packing have if one imposes the restriction that \emph{no two cylinders are parallel}. In this paper we prove two results. First, we show that there exist cylinder packings with no two cylinders parallel to each other, whose local density is arbitrarily close to the local density of a packing with parallel cylinders. Second, we construct a cylinder packing with no two cylinders parallel to each other whose global density is $1/2$. This improves the results of K. Kuperberg, C. Graf and P. Paukowisch.