10 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS
1.2.6. Periodic sphere packings. The point density
dens Λ = lim
λ→∞
card(Λ λC)
vol λC
exists in particular for periodic (point) sets
Λ =
m
i=1
ti + L,
which are given by a lattice L Rd and m translation vectors ti Rd. In this case
we simply have dens Λ m/ det L, with equality if the lattice translates ti + L are
pairwise disjoint.
By a conjecture of Zassenhaus (cf. [121]), δd is attained by periodic sets. In
Chapter 3 we study periodic sphere packings in greater detail. We in particular
consider δ as a function in a new parameter space, for periodic sets with a fixed
number of lattice translates.
One of the biggest open problems in the theory of sphere packings is to find
dimensions d with δd δd
∗,
hence dimensions, in which there exist sphere packings
which are denser than any lattice packing. There are many possible candidates
known, all them periodic packings. For example, among dimensions d 24 the
densest known sphere packings for d = 10, 11, 13, 18, 20, 22 are denser than any
known lattice packing (see [165], [259], [68], [40]).
1.3. The sphere covering problem
The sphere covering problem is, roughly speaking, concerned with the deter-
mination of most economical ways to cover
Rd
with equally sized spheres. We may
think of it as the task to choose a discrete point set in
Rd
with a given point den-
sity (average number of points in a region of volume 1), such that the maximum
distance to points of the surrounding space is as small as possible. The sphere
packing problem is sometimes described as the task to distribute a certain number
of evil dictators as far apart as possible (Tammes problem). In analogy to this, the
sphere covering problem may be described as the task to distribute a certain num-
ber of friendly governments (or supplies of some sort) such that even the furthest
“elements of the surrounding space” are as close as possible to one of them.
Although the problem is as natural as the sphere packing problem it has by
far not attracted as many attention. One reason for this are the difficulties arising
from the need to include the surrounding space into any consideration. (We learn:
For friendly governments the needs of “elements” count.) Another, at first sight
astonishing difference, is that large symmetry does not play the same role as it does
in the sphere packing problem. For example, the highly symmetric root lattice E8
is not even a “local optimum” for the lattice covering problem (see Section 5.5).
Nevertheless, the sphere covering problem is fascinating. Answers many of
them yet to be found give beautiful and exceptional subdivisions of
Rd.
Table 3
shows some recent progress to construct new currently best known sphere coverings.
So it seems, there is still much to discover and underlying principles are yet to be
found.
1.3.1. Density of thinnest sphere coverings. For a discrete set Λ and
µ R 0, we speak of a sphere covering Λ +
µBd
= {v + µx : v L, x
Bd}
if it
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