On packing spheres into containers (about Kepler's finite sphere packing problem)

TL;DR

This paper investigates the problem of packing equal spheres into minimal dilates of convex containers in Euclidean space, revealing that for large numbers of spheres, optimal packings lack simple structures like hexagonal close packings, contradicting Kepler's classical conjecture.

Contribution

It demonstrates that for smooth convex containers in dimensions two and three, optimal sphere packings cannot always be simple or regular, especially as the number of spheres grows large.

Findings

Optimal packings lack simple structure for large n.

Existence of small-radius packings that are not hexagonal close packings.

Contradiction of Kepler's conjecture in certain convex containers.

Abstract

In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container''.