What are all the best sphere packings in low dimensions?
This remarkable paper has been brewing for a long time (as has this review), partly because the questions asked are not easily formulated and partly because the results depend on certain as yet unproved postulates. The goal of the paper is to describe what may be all the tight packings (of nonoverlapping equal spheres) in dimension \(\leq 10\) and to give partial information about higher dimension. But which packings are tight. Well, those that are not loose. Suppose we can dissect the space of the packing into finitely many polyhedral pieces (possibly infinite) in such a way that the center of each center (of the spheres in the packing) lies in the interior of some piece, and there are also some empty pieces containing no centers. Then if we can rearrange the nonempty pieces into another dissection in which the centers are at least as far apart as they were originally then the packing is called (by the authors) loose. Packings that are tight in this sense certainly have the highest possible density. In 10 sections devoted in turn to dimension 1 to 10 the authors investigate tight packings. Some of the results they obtain depend on Postulate \(n\): A tight \(n\)-dimensional packing fibers over some tight \(2^k\)-dimensional packing, where \(2^k\) is the largest power of 2 strictly less than \(n\). (An \(n\)-dimensional packing \(P_n\) fibers over an \(m\)-dimensional packing \(P_m\) if \(P_n\) can be decomposed into sets -- or layers -- of points lying in parallel \(m\)-dimensional spaces, each of which is a packing of type \(P_m\).) They believe (but they cannot prove) Postulate \(n\) is true for \(2 \leq n \leq 8\); but Postulate 9 requires modification and Postulate 10 is false. In their concluding remarks they state: ``We have achieved what seems very likely to be a complete description of all the tightest packings in up to 9 dimensions, perhaps also in 10 dimensions. However, there seems to be little point in carrying these detailed arguments much further. \(\dots\) There are good reasons to believe that there are 75000 tight 25-dimensional lattice packings.'' I repeat, this is a remarkable paper.
scientific article; zbMATH DE number 764233
scientific article