Плотные упаковки шаров

How dense can we pack equal spheres in the Euclidean space? The question looks natural and is treated by humanity at least since the end of 16th century. The first four hundred years of research gave us the answers only in dimensions 1, 2, and 3. Quite recently, the answers for $N=8$ and $N=24$ - that we always presumed to be true - were proved by an elegant technique using modular forms. If we restrict ourselves to the easier situation when the centers of the spheres form a lattice the answer is known for $N$ from 1 to 8, and, of course, for $N=24$. Not too much either ... We have to ask easier questions. Can we bound the density and how? Which constructions give us packings that, if not being the best, are however dense enough? My dream is a nice theory of limit objects such as projective limits of curves or infinite extensions of ${\mathbb Q}$, as yet we are very far from it. Another great challenge is to construct lattice sphere packings that are denser that those given by a random construction (so-called Minkowski bound).