From: Dave Rusin Date: Thu, 23 Jul 1998 00:41:24 -0500 (CDT) To: mscholman@csi.com Subject: Re: How may 9" spheres could one fit into a 6' X 10' X 8' box? Newsgroups: geometry.college,sci.math In article <04f004955031578NIH2WAAF@csi.com> you write: >How may 10" spheres could one fit into a 6' X 10' X 8' box? although your subject line says >How may 9" spheres could one fit into a 6' X 10' X 8' box? This is (obviously!) the "sphere-packing" problem, and for general choices of the parameters it's not known how to find the exact answer. You can give some bounds, however. Your box measures 72" x 96" x 120", which gives it a volume of 829440 cubic inches. The volume of a sphere or radius r is (4/3)Pi(r^3), so with r=4.5 or 5 inches (you _did_ mean "10-inch _diameter_ spheres", didn't you?) we see the volume of each sphere is about 381.704 resp 523.599 cubic inches. Thus there is an upper bound of 2172 resp 1584 spheres which can fit into the box. On the other hand, we can get lower bounds by packing in a simple way. Along the 72" x 120" face we can put a 8 x 13 grid of 9" balls; ten such identical grids fit atop each other in the box, making 1040 balls. (Nine 7 x 12 grids of 10" balls fit similarly, making 756 larger balls). You can improve these lower bounds by alternating the given grids with smaller grids nestled into the crevices of the first: alternate a 8 x 13 grid with a 7 x 12 grid. Better yet, arrange the balls on each level into a honeycomb pattern (8 rows of 8 with 7 rows of 7 between them) and then stagger such grids in each level; this is known to allow you to fill about 3/4 of the available space with spheres (I didn't check for rounding conditions, but this should get you about 1600 9" balls, and about 1150 10" balls into the box). It is not known whether the hexagonal arrangements I described give the optimal limiting density for large boxes; this is thought to be true, and a proof has been announced by Hsiang, but has not been presented in detail and has been greeted with a certain skepticism. In practice, if I were you, I'd follow the wisdom of fruit vendors and pack the balls as indicated! dave