05-18
On packing spheres into containers (about Kepler\'s finite sphere packing problem)
Preprint series: 05-18, Preprints
- MSC:
- 52C17 Packing and covering in $n$ dimensions, See also {05B40, 11H31}
- 05B40 Packing and covering, See also {11H31, 52C15, 52C17}
- 01A45 17th century
Abstract: In Euclidean $d$-spaces, 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 sequences of solutions to the container problem can not have a ``simple structure\'\'. By this we in particular find that there exist arbitrary small $r>0$ such that packings with spheres of radius $r$ into a smooth $3$-dimensional convex body 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\'\'.
Keywords: sphere packings, Kepler, container problem
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