From: asimov@nas.nasa.gov (Daniel A. Asimov) Newsgroups: sci.math.research Subject: Re: Densest Spherical Packing for Dimension > 2? Date: 8 Sep 1995 17:49:09 GMT >>for long time the densest spherical packing has only been definitely >>known for dimension <= 2 [1]. It has been assumed that the >>3-dimensional hexagonal packing (hcp, and its counterpart, fcc, for >>the crystallographs :) *is* indeed the densest spherical packing for >>dimension 3. it was stated that >Conway proved something to the effect that it was the best periodic packing, >I think. Oh, your reference is probably to that result. -------------------------------------------------------------- I believe it was Gauss who showed -- in the early 1800s -- that the fcc lattice in R^3 is the densest 3D sphere packing among lattice packings. I don't know if Gauss published this, but apparently this result is included in the paper by Korkine and Zolotareff, Sur les formes quadratiques, Math. Ann. v. 6 (1873), 366-389. (Cf. Conway and Sloane, What are all the best sphere packings in low dimensions, Discrete Comput. Geom., v. 13 (1995), 383-403.) --Dan Asimov ============================================================================== From: kuperberg-greg@MATH.YALE.EDU (Greg Kuperberg) Newsgroups: sci.math.research Subject: Re: Covering problem - reference please? Date: 9 Sep 1995 16:13:21 -0400 In article WimmerLienha@edvz.sbg.ac.at (your_name) writes: > >This problem seems to be connected with the following: place n congruent >circles on the surface of a sphere, so that they cover the surface, and that >their radius is minimal. This problem was treated by L.Fejes-Toth [1,2], who >proofed an inequality, giving solutions for n = 3,4,6,12; G.Fejes-Toth [3] >solved it for n = 10 and 14, and T.Tarnai & Zs.Gaspar [4] found a way to >compute good point sets for n up to 20, but without any proof. This happens to be related to the Hsiang controversy. The L. Fejes-Toth bound is a bound for arbtirary r and n on the fraction of the 2-sphere that can be covered by n disks of radius r. When n=3,4,6, or 12, the bound is optimal for all r. It therefore simultaneously establishes the densest packing, the thinnest covering, and the smallest circumscribed n-sided polyhedron for these four values of n. In particular, it establishes that the smallest dodecahedron containing the sphere is regular, which is closely related to the still-open conjecture that the regular dodecahedron is also the smallest possible Voronoi region in a sphere packing. (It's smaller than the Voronoi regions of the fcc and hcp packings.) As a warm-up to the sphere packing conjecture, Hsiang claimed a solution to the Voronoi region conjecture. His argument is apparently based on an alternate proof of the smallest dodecahedron theorem. (He did not at first know that the latter was solved.) But the consensus among people like the Fejes Toths is that the dodecahedron theorem is deceptively weak as a partial result towards the Voronoi region conjecture. Hsiang's attempt to bridge the gap has the same weaknesses as his claimed proof of the much harder sphere packing conjecture. ============================================================================== From: ctm@riga.berkeley.edu (C. T. McMullen) Newsgroups: sci.math.research Subject: Re: s.m.r. Problems for the next century Date: 12 Apr 1997 14:55:45 GMT [deletia -- djr] Hsiang's proof (or "strategy of proof", as it is called in Math Reviews) of the Kepler conjecture appeared in Internat. J. Math. 4 (1993).