On packing spheres into containers

  • Achill Schürmann

    Mathematics Department University of Magdeburg 39106 Magdeburg Germany
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In an Euclidean dd-space, the container problem asks to pack nn equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and d2d\geq 2 we show that solutions to the container problem can not have a "simple structure" for large nn. By this we in particular find that there exist arbitrary small r>0r>0, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius rr, 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".

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Achill Schürmann, On packing spheres into containers. Doc. Math. 11 (2006), pp. 393–406

DOI 10.4171/DM/215