We consider now a specific class of almost periodic sphere packings.
These packings contain also periodic packings but the computations do not
rely on the distinction between periodic and aperiodic. In order to find
dense packings experimentally, it was sometimes
of advantage, not to distinguish between
periodic and aperiodic packings.
Take a finite union of disjoint half-open intervals , a
rotation vector , a radius r bigger than zero and the standard basis
Define , where is the Euclidean -norm in . If , we get for any a sphere-packing
If we put a sphere of radius r/2 at each point of H, we get a r-packing because the maximal distance between two different points in H is by construction ;SPMlt;r. This packing is not periodic if one of the numbers is irrational. We call such a packing quasi-periodic.
The density of a quasi-periodic sphere-packing can be approximated explicitly by periodic sphere-packings because every vector is a limit of rational vectors and the density depends continuously on . It is not excluded that in some dimension, the highest possible packing is aperiodic, but we will see that if J consists of a single interval, the maximal density is obtained by a periodic packing.
The center density of a sphere-packing is defined as , where is the density and V(d) is the volume of the unit sphere in .
Remark. Note that evenso the map is not continuous, the density is continuous.
|Fig. 3. The set with the vector . On the right, we see the set and the interval .|
Given , we find J consisting of one interval such that . Start with and form the finite set
The points nearest to are symmetric to 0.
Form J=[0,c). We check that .
The center density of the packing with this interval is then .
1) We can add a second interval J' as follows. Assume the interval of length is the largest interval disjoint from . There are three cases. If , take . If , define . In the other case, take . We show now that, . For , we have and , since J' is disjoint from . For , we have also that is disjoint from J'. It is also disjoint from , since . The center density of the packing is then .
2) We could proceed as follows to get larger and larger sets J. Given , take any point outside . If is disjoint from , we can find a maximal interval which has the property . Form . Repeating this construction leads to a finite or countable union J of half-open intervals which has the property that . In other words, we are adding spheres until the covering is saturated, in the sense that there is no longer room with positive density for adding additional spheres. Usually, in our experiments if the first interval is large enough, then J is already saturated and no second interval is needed.
3) For getting high densities, we have to choose r such that the number of lattice points inside the ball of radius r is just below a point of discontinuity. Necessary is , where and that there is a lattice point on the boundary of the ball with radius r so that a further increase of r increases the number of lattice points. By a theorem of Lagrange in number theory, this is always true if and . In other words, by extending the variational problem and varying r also, a packing which maximizes the density has the property that is an integer.
4) We expected to get high densities for radii r, for which many lattice points are on the boundary of a sphere of radius r because of a possible large kissing number. The experiments confirm this in some cases like r=5, d=2,3, but it was not the rule.
|Fig. 4. The piecewise linear function in the case r=5. As larger the value, as brighter is the point . On the black resonance lines, the value is zero.|
The same argument shows that for any finite set not containing the origin, the function
takes its global maximum on a rational point . We can consider therefore any packing problem, where the spheres are replaced by some compact set . The analogues problem is to find for a given r, the densest packing of with copies of . The same construction gives a d-dimensional manifold of almost periodic packings for which the density exists. The maximal density on this manifold is obtained by periodic packings.