Packing

2020 Mathematics Subject Classification: Primary: 52C20 Secondary: 52C22 [MSN][ZBL] \( \def \Z { {\cal Z}} \)

A packing of a (finite or infinite) family of sets $M_i$ in a set $A$ is, in its strict sense, any pairwise disjoint family of subsets $M_i\subset A$.

However, in geometry, for packings in Euclidean spaces, on an $n$-dimensional sphere, in a closed domanin, or in some other manifold, the sets $M_i$ are often closed domains, and then this condition is relaxed to requiring only that the interiors of the sets are pairwise disjoint.

Lattice packings

As a special case, in vector spaces $V$, such as $\R^d$, packings of translates $ \{ M+v \mid v \in \Z \} $, of a set $ M \subset V $ (also called the packing $(M,\Z)$ of $M$ by $\Z$) are considered. If the set $ \Z \subset V $ of translation vectors is a point lattice, then the packing is called a lattice packing. In particular, such packings are investigated in the geometry of numbers and in discrete geometry.

Sphere packings

Packings of congruent spheres are considered both in the geometry of numbers and in discrete geometry, and have applications in coding theory. A central problem is finding the densest packing, and the densest lattice packing, of congruent spheres in $\R^d$. For $d=3$, the problem (known as Kepler conjecture or Kepler problem) to decide whether there is a better packing than the densest lattice packing was a famous open problem that was recently solved by Hales (1998). With the help of massive computer calculations he showed that the densest lattice packing of spheres is optimal. This result is generally considered as correct but because of its size it has not yet been verified independently. However, Hales and his team are working on a computer-verifyable version of the proof.

Tilings

A tiling is a packing without gaps, i.e., such that the $M_i$ are also a covering of $A$.

References

[Ba]	E.P. Baranovskiĭ, "Packings, coverings, partitionings and certain other arrangements in spaces with constant curvature" (1969) Algebra. Topology. Geometry. (1967) (Russian) pp. 189–225 Akad. Nauk SSSR Vsesojuz. Inst. Naučn. Tehn. Informacii, Moscow MR0257885
[CoSl]	J.H. Conway, N.J.A. Sloane, "Sphere packing, lattices and groups", Springer (1988) MR0920369
[FeTo]	L. Fejes Tóth, "Lagerungen in der Ebene, auf der Kugel und im Raum", Springer (1972) MR0353117 Zbl 0229.52009
[Ro]	C.A. Rogers, "Packing and covering", Cambridge Univ. Press (1964) MR0172183 Zbl 0176.51401