The figure shows the piecewise linear function
(a1,a2) -> N(2,r,a) which is the minimum of all (n,a) mod 1, where
n runs over all nonzero integer lattice points n satisfying ||n|| smaller
than r. The picture shows the case of packings with spheres of radius r=5
in dimension d=2. As larger the packing density, as brighter is the point
(a1,a1) on the two dimensional torus parameterizing the packings.
On the orange resonance lines, the packing density is zero.
We proved that in all dimensions d and all radii r, the function
a -> N(d,r,a) has only rational "critical points".
This means that on the corresponding finite
dimensional manifolds of almost periodic packings,
the maximal density is achieved by periodic packings.