On Translative Packing Densities in $E^2$ and $E^3$

Abstract

The theory of packing and covering is an essential part of discrete geometry. In this dissertation we focus on and contribute to the knowledge on the densities of translative and lattice packings in $E^{2}$ and $E^{3}$. $d_{T}(C)$ and $d_{L}(C)$ will be used to denote the largest translative packing density and the largest lattice packing density of a planar disc or three dimensional body $C$, and for short, we will call them the translative packing density and the lattice packing density, respectively.

In 1892, Thue solved the problem of the densest packing of congruent circular discs in the plane. In 1950s, Rogers proved that for any convex disc $C$, $d_{T}(C) = d_{L}(C)$. This result was generalized by L. Fejes T\'{o}th in 1985 to limited semi-convex domains. Besides, Fejes T\'{o}th posed the question whether Rogers's equality remains true for non-convex domains. A. Bezdek answered this question negatively by providing a nonconvex disc, resembling a wrench. Bezdek determined the lattice packing density of his wrench and showed a non lattice-like translative packing of the wrench with a larger density. Note that Bezdek did not have to prove that the later packing has the largest density among translative packings, and this is the point where I joined this research area and proved the following:

A. First, I proved what Bezdek already conjectured. Specifically, I showed that the translative packing, which Bezdek included in his paper, is in fact a densest translative packing of his wrench.

B. Once A) was proved I could complete a new proof of Bezdek’s result. This time all I had to prove was that lattice packings of the wrench cannot have a density equal to the largest translative packing density of the wrench.

C. As a preparation for studying lattice packings in $E^3$, I proved a geometric property of point lattices. The one I proved could be interesting on its own. Let us assume that a point lattice contains all points whose position vectors are integer linear combinations of three independent vectors. One cannot expect that $8$ of these lattice points form the vertices of a cube whose faces are parallel to coordinate planes. But for every $\varepsilon$, we can guarantee the existence of 8 lattice points which are vertices of a large parallelepiped, so that after proper scaling it is in the $\varepsilon$-neighborhood of a unit cube. We call such parallelepipeds $\varepsilon$-cubes.

D. It would be interesting to explore translative packing densities in $E^3$, so I revisited Rogers's equality $d_{T}(C) = d_{L}(C)$, where $C$ denotes a convex disc in the plane. The question whether the same equality holds in $E^{3}$ is still open today for convex bodies. I proved that the equality holds for cylinders with convex base.

E. Naturally, we would like to determine the largest translative packing density of cylinders whose base is Bezdek’s wrench (called 3D-wrench). It was conjectured that stacking 3D-wrenches vertically over the densest planar lattice will give the densest 3D lattice. Surprisingly, this was not the case. It turned out that there is a different lattice packing of a single 3D-wrench, whose density is equal to the 3D-wrench's translative packing density.