Packings of Spheres in Cylindrical Confinement

The quest to densely pack spheres in unbounded space has its roots in Kepler’s Conjecture (1611), which proposed a hexagonal close-packed configuration as the most efficient arrangement in an infinite three-dimensional space. While such studies, including those for packing in higher dimensions, have illuminated the densest bulk arrangements, in recent years the problem of packing in confined spaces has also gained significant interest. An example is the densest packing of spheres within infinitely long cylindrical channels, with optimal (densest) configurations determined by the ratio D/d of the cylinder diameter D to the sphere diameter d. Over the last few decades, the study of packing within cylinders has undergone notable advancements. Beyond its foundational scope, it now covers a range of themes, yielding a number of experimental discoveries. This chapter endeavours to chart the history of this subject, including its intersections with phyllotaxis (a branch of biology dedicated to examining the patterns and arrangements of stems and leaves in plants, which has a special significance here). In this chapter, I aim to provide an overview of this developing field, spotlighting some of the latest developments and some open questions.