14 1. FROM QUADRATIC FORMS TO SPHERE PACKINGS AND COVERINGS

Figure 7. A close sphere packing given by the hexagonal lattice.

d lattice γd authors

2 A2

∗

2/

√

3 ≈ 1.154700 Ryshkov [211]

3 A3

∗

5/3 ≈ 1.290994 B¨ or¨ oczky [44]

4 Ho4 2

√

3(

√

3 − 1) ≈ 1.362500 Horv´ ath [139]

5 Ho5 3/2 +

√

13/6 ≈ 1.449456 Horv´ ath [140]

Table 4. Known optimal lattice packing-coverings.

raised by L. Fejes T´ oth [109] of finding “close packings” attaining this gap-radius

is another formulation of the packing-covering problem.

The last interpretation shows that γd ≥ 2 would imply that in any d-dimensional

lattice packing with spheres of unit radius there is still space for spheres of radius

1. Hence, δd ∗ ≥ 2δd, which implies of course in particular, that densest sphere

packings in dimension d are non-lattice packings. Thus the open problem of the

existence of d with γd ≥ 2 is a very challenging question, in particular in view of

the asymptotic bound γd ≤ 2 + o(1) due to Butler [50].

As for the lattice covering problem, the lattice packing-covering problem has

been solved up to dimension 5 (see Table 4). Ryshkov [211] solved the general

2-dimensional case. The 3-dimensional case was settled by B¨ or¨ oczky [44], even

without the restriction to lattices. The 4- and 5-dimensional lattice cases were

solved by Horv´ ath [139], [140]. Note that the lattices Ho4 and Ho5 (see Section 5.3)

discovered by Horv´ ath are neither best covering, nor best packing lattices. The

results were obtained by using Voronoi’s reduction theory, which is described in

Section 4.1. So far it is an open problem whether or not γd

∗

γd for any d ≥ 4.

Our computations, described in Section 5.3, verify all of the known lattice

results in dimensions ≤ 5. As for the covering problem, none of the values γd has

been determined in a dimension d ≥ 6 so far. In Section 5.3 we report on new best

known packing-covering lattices for d = 6, 7. We thereby show in particular that

γ6 1.412, revealing the phenomenon γ6 γ5, suspected by Lagarias and Pleasants

in [160], and observed by Zong [272], who showed that γ(E6)

∗

=

√

2 γ5.

We were not able yet to find any new best known lattices in dimensions d ≥ 8.

Nevertheless, because of their symmetry and the known bounds on γd, Zong [272,

Conjecture 3.1] made the following conjectures: E8 and Leech lattice Λ24 are