Maximizing the packing density on a class of almost periodic sphere packings 
Oliver Knill, August 29, 1995
This paper appeared in Expositiones Mathematicae, 14 (1996), p. 227246, 1996 MR 97i:52022 
Abstract 
We consider the variational problem of maximizing the packing density on some finite dimensional set of almost periodic sphere packings. We show that the maximal density on this manifold is obtained by periodic packings. Since the density is a continuous, but a nondifferentiable function on this manifold, the variational problem is related to number theoretical questions. Every sphere packing in defines a dynamical system with time . If the dynamical system is strictly ergodic, the packing has a well defined density. The packings considered here belong to quasiperiodic dynamical systems, strictly ergodic translations on a compact topological group and are higher dimensional versions of circle sequences in one dimension. In most cases, these packings are quasicrystals because the dynamics has dense point spectrum. Attached to each quasiperiodic spherepacking is a periodic or aperiodic Voronoi tiling of by finitely many types of polytopes. Most of the tilings belonging to the ddimensional set of packings are aperiodic. We construct a oneparameter family of dynamically isospectral quasiperiodic sphere packings which have a uniform lower bound on the density when varying the radius of the packing. The simultaneous density bound depends on constants in the theory of simultaneous Diophantine approximation. 

Two Pascal programs (with C translation), which I wrote for this article.
