The packing question for spheres has been formulated by Hilbert
for general compact subsets K
of . An example is to replace the Eucledean norm by the norm
and to pack with with
spheres and clearly for p=1 or , the maximal
packing density is 1 and different p will probably lead to different
lattices with optimal density. Parameters p, where the optimal packing type
changes should be considered as a bifurcation parameter.
For general K, the construction of quasiperiodic packings is the same as already described. We have only to replace by to obtain the corresponding densities. General packing problems with different type of subsets K are interesting and relevant in crystallography or chemistry, where K takes the shape of a molecule. The same proof as shown above shows that the maximal packing density on d-dimensional manifolds of quasi-periodic packings is attained by periodic packings if we take only copies of K which are obtained from K by translation by . Removing the last restriction might lead in general to aperiodic packings with maximal density: an indication is that there exist convex sets in , the Schmitt-Conway-Danzer tiles, for which the densest packing is aperiodic with density 1. We mention also a result in  which says that for a generic set of convex sets in the plane, the densest packing is not lattice like.
Crystals in nature are often periodic. A situation closest
to the periodic case are quasi-periodic crystals which are a family of
so called quasicrystals. Our result that in
a class of quasi-periodic crystals, the maximal density is attained on
periodic crystals can have physical relevance since high
densities are favored by stable crystalline structures.
If the packing is almost periodic, there are large domains, where the packing
looks periodic, but there must be defects, if the lattice is not
For , consider the -collar of
the packing S which consists of all points in which have distance
from any ball. There exists a largest ,
such that the complement of still percolates.
The defect, the complement of with critical
is not necessarily almost periodic since the
topology of does not depend continuously on .
Moving on the manifold of almost periodic rational r-packings parameterized
by corresponds to (non continuous)
deformations of the crystal. The density, when considered as
a negative "energy" of the crystal is a piecewise linear continuous
function on the parameters. Variational methods do not apply in order
to find the maximal density which correspond to minimizing the energy
because the energy is not differentiable at the maximum. However, we have
seen that the minimal energy on this manifold is obtained by periodic
crystals. Since atomic dense packings are believed to be fundamental in
condensed matter, our toy model can in principle explain an aspect of
the stability of crystals. Moreover, the periodicity of some of the
crystals can be so large that it would be hard to distinguish them
experimentally from a true quasicrystal.
Aperiodic Voronoi tilings.
For strictly ergodic packings,
there exists a constant C such that if then there exists
with some m satisfying . In other words, there
are only finitely many sphere-configurations which can occur in a ball of
The Voronoi tiling of is built up by
finitely many Voronoi simplices. This Voronoi decomposition is an
aperiodic tiling of if rationally independent.
The diffraction spectrum of the quasicrystals considered here can
be computed explicitly: it is a point measure with weights given
by a Fourier transform of (see ).
Random sphere packings. We should compare the quasi-periodic packings in with random sphere packings, which are interesting from the practical point of view. In three dimensions, computer simulations and experiments with large containers of steel balls give a density of 0.6366 (see ). Higher densities are no more believed to be random, since the high density is then obtained by crystalisation. Almost periodic sphere packings in three dimensions with higher densities are obtained for most radii r.
Generalization of the construction.
The construction could be generalized in the following way.
Take any compact abelian group G (generalizing )
and d commuting group translations .
Consider a ring of measurable sets
(generalizing the ring of half open intervals), where
each element in different from has positive Haar
measure and such that the boundary of every has zero
Take with maximal measure such that
and do the same construction as before.
We get a strictly ergodic subshift defining the centers of a sphere packing
with spheres of radius r/2 and the
packings define quasicrystals with pure point dynamical spectrum and so
pure point diffraction spectrum.
We think however that no other group is as convenient as .
Taking the p-adic group would be a natural choice from the dynamical
systems point of view.
A replacement of the irrational rotation with d commuting homeomorphisms of the circle does not lead to more general packings: if one of the rotation numbers of these homeomorphisms is irrational, then this - action is topologically conjugated to the action given by the irrational rotations. The subtle and still unsolved problem, whether d commuting circle diffeomorphisms are smoothly conjugated to irrational rotations is not involved.