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

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 $d\ge 2$ 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''.

13 pages, 2 figures; v2: major revision, extended result, simplified and clarified proof

References in corpus (1)

• Lectures on 0/1-polytopes

Cited by in corpus (1)

• Upper bound on the disordered density of sphere packing and the Kepler Conjecture