# Periodicity of Ideal Packings on Arbitrary Boundaries

This is a question that's been nagging me for some time that I find particularly interesting, but have no idea how to go about solving it.

Consider some arbitrary solid $S$, such that $\partial S$ is topologically a sphere, and consider a second arbitrary solid $D$, with the same requirement.

We define the 'ideal packing' within some solid (in our case $S$) to be the packing that minimizes the amount of volume not contained by a set of our packed solids (whose structure is $D$) with the additional requirement that our packed solids do not overlap and do not have any points outside of the containing solid, $S$.

My question then would be: is it possible to prove that the ideal packing has to be periodic [probably not true at the boundaries]? Is this true at least locally as we increase the number of packed solids? Is it true at all?