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

by Achill Schürmann.

Series: 2005-18, Preprints

52C17 Packing and covering in $n$ dimensions
05B40 Packing and covering
01A45 17th century

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''.

sphere packings, Kepler, container problem