On packing spheres into containers (about Kepler's finite sphere packing problem)

In an Euclidean $d$-space, the container problem asks to pack $n$ equally sized spheres into a minimal dilate of a fixed container. If the container is a smooth convex body and $d\geq 2$ we show that solutions to the container problem can not have a ``simple structure'' for large $n$. By this we in particular find that there exist arbitrary small $r>0$, such that packings in a smooth, 3-dimensional convex body, with a maximum number of spheres of radius $r$, are necessarily not hexagonal close packings. This contradicts Kepler's famous statement that the cubic or hexagonal close packing ``will be the tightest possible, so that in no other arrangement more spheres could be packed into the same container''.Comment: 13 pages, 2 figures; v2: major revision, extended result, simplified and clarified proo

Computational Approaches to Lattice Packing and Covering Problems

We describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z^d. In practice, our implementations reproduce known results in dimensions d <= 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.Comment: (v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in Discrete and Computational Geometry, see also http://fma2.math.uni-magdeburg.de/~latgeo

Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8

We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the formerly best known given by the lattice A8*. Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense.Comment: 13 pages; (v2) major revision: proof of rigidity corrected, full discussion of E8-case included, src of (v3) contains MAGMA program, (v4) some correction

Classification of eight dimensional perfect forms

In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.Comment: 14 page

Inhomogeneous extreme forms

G.F. Voronoi (1868-1908) wrote two memoirs in which he describes two reduction theories for lattices, well-suited for sphere packing and covering problems. In his first memoir a characterization of locally most economic packings is given, but a corresponding result for coverings has been missing. In this paper we bridge the two classical memoirs. By looking at the covering problem from a different perspective, we discover the missing analogue. Instead of trying to find lattices giving economical coverings we consider lattices giving, at least locally, very uneconomical ones. We classify local covering maxima up to dimension 6 and prove their existence in all dimensions beyond. New phenomena arise: Many highly symmetric lattices turn out to give uneconomical coverings; the covering density function is not a topological Morse function. Both phenomena are in sharp contrast to the packing problem.Comment: 22 pages, revision based on suggestions by referee, accepted in Annales de l'Institut Fourie

A generalization of Voronoi's reduction theory and its application

We consider Voronoi's reduction theory of positive definite quadratic forms which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more general, the theory is developed for forms which are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real thin algebraic number fields which was recently initiated by Bayer-Fluckiger and Nebe. Moreover, we apply it to construct new best known sphere coverings in dimensions 9,..., 15.Comment: 31 pages, 2 figures, 2 tables, (v4) minor changes, to appear in Duke Math.

Point configurations that are asymmetric yet balanced

A configuration of particles confined to a sphere is balanced if it is in equilibrium under all force laws (that act between pairs of points with strength given by a fixed function of distance). It is straightforward to show that every sufficiently symmetrical configuration is balanced, but the converse is far from obvious. In 1957 Leech completely classified the balanced configurations in R^3, and his classification is equivalent to the converse for R^3. In this paper we disprove the converse in high dimensions. We construct several counterexamples, including one with trivial symmetry group.Comment: 10 page

Complexity and algorithms for computing Voronoi cells of lattices

In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.Comment: 20 pages, 2 figures, 5 table