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Crystallographic considerations. The packing question for spheres has been formulated by Hilbert for general compact subsets K of tex2html_wrap_inline555 . An example is to replace the Eucledean norm by the norm and to pack tex2html_wrap_inline555 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 tex2html_wrap_inline1359 by tex2html_wrap_inline1779 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 tex2html_wrap_inline1357 . Removing the last restriction might lead in general to aperiodic packings with maximal density: an indication is that there exist convex sets in tex2html_wrap_inline565 , the Schmitt-Conway-Danzer tiles, for which the densest packing is aperiodic with density 1. We mention also a result in [15] which says that for a generic set of convex sets in the plane, the densest packing is not lattice like.

Quasicrystals. 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.

Defects. 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 periodic. For tex2html_wrap_inline803 , consider the tex2html_wrap_inline623 -collar tex2html_wrap_inline1801 of the packing S which consists of all points in tex2html_wrap_inline555 which have distance tex2html_wrap_inline1807 from any ball. There exists a largest tex2html_wrap_inline803 , such that the complement of tex2html_wrap_inline1801 still percolates. The defect, the complement of tex2html_wrap_inline1801 with critical tex2html_wrap_inline623 is not necessarily almost periodic since the topology of tex2html_wrap_inline1801 does not depend continuously on tex2html_wrap_inline623 .

Minimizing energy. Moving on the manifold of almost periodic rational r-packings parameterized by tex2html_wrap_inline1823 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 tex2html_wrap_inline1827 then there exists tex2html_wrap_inline1829 with some m satisfying . In other words, there are only finitely many sphere-configurations which can occur in a ball of given radius. The Voronoi tiling of tex2html_wrap_inline555 is built up by finitely many Voronoi simplices. This Voronoi decomposition is an aperiodic tiling of if tex2html_wrap_inline791 rationally independent.

Diffraction spectrum. 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 [10]).

Random sphere packings. We should compare the quasi-periodic packings in tex2html_wrap_inline565 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 [18]). 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 tex2html_wrap_inline1851 ) and d commuting group translations tex2html_wrap_inline1855 . Consider a ring tex2html_wrap_inline1857 of measurable sets (generalizing the ring of half open intervals), where each element in tex2html_wrap_inline1857 different from has positive Haar measure and such that the boundary of every tex2html_wrap_inline1861 has zero Haar measure.

Take tex2html_wrap_inline1861 with maximal measure such that tex2html_wrap_inline1865 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 tex2html_wrap_inline1851 . 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 tex2html_wrap_inline587 - 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.

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Next: Questions Up: Maximizing the packing density Previous: Coverings

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