Sphere packings

In a paper in Expos. Math. 14, 227-246 (HTML version) a class of quasiperiodic sphere packings was studied. My hope was to find dense packings (in any dimension) by allowing the packing to be a quasi crystal. Among the class of packings, which were considered, periodic packings give the highest density. Computer assisted searches on those manifolds of quasiperiodic packings allows to construct many packings with explicitely known density.

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.

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