From: Robin Chapman Subject: Re: Sphere packings in R^n Date: Mon, 19 Apr 1999 08:54:12 +1000 Newsgroups: sci.math To: Toby Kelsey Keywords: optimal lattice packings Toby Kelsey wrote: > > The "hexagonal close packed" lattice for R^n has n(n+1) nearest > neighbours. Are there generally denser packings, or is that the best? > This is optimum only for n <= 3. It's been known for a long time that hexagonal packing is optimum in 2 dimensions and Thomas Hales has recently announced a proof in three dimensions. See http://www.math.lsa.umich.edu/~hales/countdown/ . The optimum *lattice* packings are known in dimensions up to 8. For dimensions 4 to 8 they are called the D_4, D_5, E_6, E_7, E_8 packings. The D_n packing is basically a chessboard packing. One takes spheres centred at (a_1, ... ,a_n) where the integers are a_j have even sum. Then one can fit spheres with diameter sqrt(2) together with centres at these points. There are 2n(n-1) neighbouring spheres to the central one. The E_6, E_7 and E_8 packings have 72, 126 and 240 neighbours respectively. The E_8 packing is especially interesting. It's twice the density of D_8 and obtained from D_8 by adding spheres with centres at (a_1/2, ..., a_8/2) where the a_j s are odd integers and a_1 + ... + a_8 is even. In dimensions > 8 the optimum lattice packings are not known. There are some dimensions where non-lattice packings are known which are denser than any known lattice packing. In 24 dimensions there is the most fascinating example: the Leech lattice with 196560 spheres touching a given sphere. Its symmetry group is essentially the first Conway simple group. The bible on sphere packings is Conway and Sloane, "Sphere Packings, Lattices and Groups" published by Springer-Verlag. -- Robin Chapman + "Going to the chemist in Department of Mathematics, DICS - Australia can be more Macquarie University + exciting than going to NSW 2109, Australia - a nightclub in Wales." rchapman@mpce.mq.edu.au + Howard Jacobson, http://www.maths.ex.ac.uk/~rjc/rjc.html - In the Land of Oz