Geodesic
On packing spheres into containers
Documenta mathematica, Tome 11 (2006), pp. 393-406

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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≥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”.
DOI : 10.4171/dm/215
Classification : 01A45, 05B40, 52C17
Mots-clés : sphere packing, Kepler, container problem 
Achill Schürmann. On packing spheres into containers. Documenta mathematica, Tome 11 (2006), pp. 393-406. doi: 10.4171/dm/215
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