Finding sphere packings in with maximal density
are important mostly unsolved
problems with many applications (see ).
The problem is interesting in
geometry, group theory, crystallography or physics.
Here, we look at the problem from a dynamical system point
|It is still controversial, whether the Kepler conjecture claiming that there is no sphere-packing in with density higher than 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.|
Periodic packings and often lattice packings
have achieved the highest today known densities (see ).
Following an advise in  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
which form a part of the huge space 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
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 . 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 can be described by a configuration
The closure Y of all the translates of x in
forms a subshift with time .
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 ).
|Fig. 1 Part of a two dimensional aperiodic packing with spheres of radius 5 having centers located on a subset of . This packing is close to the best existing packing in two dimensions.|
|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 .|