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 $d\geq 2$ 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''.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- June 2005
- DOI:
- 10.48550/arXiv.math/0506200
- arXiv:
- arXiv:math/0506200
- Bibcode:
- 2005math......6200S
- Keywords:
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- Mathematics - Metric Geometry;
- 52C17 (Primary) 05B40;
- 01A45 (Secondary)
- E-Print:
- 13 pages, 2 figures