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Introduction

Finding sphere packings in tex2html_wrap_inline555 with maximal density are important mostly unsolved problems with many applications (see [3]). The problem is interesting in geometry, group theory, crystallography or physics. Here, we look at the problem from a dynamical system point of view.

It is still controversial, whether the Kepler conjecture claiming that there is no sphere-packing in tex2html_wrap_inline565 with density higher than tex2html_wrap_inline567 is open or proven ([8, 11]).
Note (added in June 1998). It seems that now the majority of the experts have the opinion that Kepler conjecture is still open.
We cite Gabor Fejes Toth in a Review article on Hsiang's proof
"If I am asked whether the paper fulfills what it promises in its title, namely a proof of Kepler's conjecture, my answer is: no".
A promising program to prove the conjecture is persued by T. Hales. See T. Hales, Sphere packings. I. Discrete Comput. Geom. 17 (1997), 1-51, and Sphere packings. II. Discrete Comput. Geom. 18 (1997), 135-149.
Also for dimensions bigger than 3, the best densities are still not known evenso there is a list of candidates (see [2],[3]). According to a general result of Groemer ([6] see also [7]), there exists a sphere packing which has a maximal density.

Periodic packings and often lattice packings have achieved the highest today known densities (see [3]). Following an advise in [16] p. 188, we restrict us here to a subclass of packings, where we know that the density of the packing exists. This family of quasicrystals, almost periodic sphere-packings in tex2html_wrap_inline555 which form a part of the huge space tex2html_wrap_inline573 of all packings. Quasicrystalline dense sphere packings are of interest in physics (see for example [14, 1]). The ergodic-theoretical side of the problem comes natural in the context of aperiodic tilings. Tilings and packings are closely related since to every strictly ergodic sphere-packing is attached a tiling of tex2html_wrap_inline555 by Voronoi cells consisting of finitely many types of polytopes.
The quasicrystals in this article are defined by higher dimensional circle sequences and have the property that the densities can be computed explicitly from the parameters. This allows for a machine to search through many of these packings and to determine in each case the exact density. The feature of being able to compute macroscopic quantities like densities for aperiodic configurations was also useful when we studied cellular automata on circle configurations [10]. The main motivation for the present work was to investigate dense packings with a new class of packings. We could consider many packings with high periodicity which are close to the best known packing.

A packing tex2html_wrap_inline577 can be described by a configuration tex2html_wrap_inline579 . The closure Y of all the translates of x in tex2html_wrap_inline585 forms a subshift with time tex2html_wrap_inline587 . In general, there exist many shift- invariant measures on Y and the density depends on the choice of the invariant measure. For the packings, we consider here, the subshift Y has a unique shift-invariant measure and it implies that the density is well defined and can explicitly be determined. Moreover, if we make a translation in the i'th coordinate, we get a one-dimensional strictly ergodic subshift for which the dynamical spectrum (and so the diffraction spectrum) is known to be pure point. The packings are therefore crystals or quasicrystals (see [13]).

tex2html_wrap605 Fig. 1 Part of a two dimensional aperiodic packing with spheres of radius 5 having centers located on a subset of tex2html_wrap_inline597 . This packing is close to the best existing packing in two dimensions.
The organization of this article is as follows. In section 2, we relate sphere packings with dynamical systems and point out, that the uniquely ergodic packings have a well defined density. In section 3, we consider a finite dimensional class of strictly ergodic sphere packings and compute the density of the packings. We show that the density functional takes its maximum on periodic packings. In section 4, we give classes of such packings and relate the problem to find dense packings to number theoretical questions. We also report about packings found by numerical experiments. In section 5, we will construct a one-parameter family of isospectral sphere packings which have a uniform positive density depending on constants in simultaneous Diophantine approximation. In section 6, we discuss a related construction for coverings. Relations with other topics are outlined in section 7 and in section 8, some open problems are summarized.
tex2html_wrap607 Fig. 2. Part of a three dimensional aperiodic packing with density 0.717. The spheres have radius r=5 and are located on a subset of tex2html_wrap_inline603 .

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Next: Sphere packings and dynamical Up: Maximizing the packing density Previous: Maximizing the packing density

Oliver Knill
Mon Jun 22 17:57:55 CDT 1998