On non-parallel cylinder packings

TL;DR

This paper investigates the density bounds of non-parallel cylinder packings in three-dimensional space, proving a conjecture about maximum density and establishing a lower density bound close to a specific value.

Contribution

It proves Kuperberg's conjecture on the maximum upper density and shows the existence of non-parallel cylinder packings with high lower density.

Findings

Proof of Kuperberg's conjecture for upper density ${rac{ extpi}{ extsqrt{12}}}$

Existence of non-parallel cylinder packings with lower density arbitrarily close to ${rac{ extpi}{6}}$

Advances understanding of optimal packing densities in 3D geometry

Abstract

In this paper we will discuss optimal lower and upper density of non-parallel cylinder packings in $R^{3}$ and similar problems. The main result of the paper is a proof of the conjecture of K. Kuperberg for upper density (existence of a non-parallel cylinder packing with upper density ${\pi}/{\sqrt{12}}$). Moreover, we prove that for every $\varepsilon > 0$ there exists a nonparallel cylinder packing with lower density greater then ${\pi}/{6} - \varepsilon$.