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