-----Original Message----- From: John Conway [mailto:conway@Math.Princeton.EDU] Sent: Monday, September 29, 2003 3:34 PM To: math-fun Subject: Re: [math-fun] More on sphere packing

On Mon, 29 Sep 2003, Dan Hoey wrote: I didn't read the piece I snipped, but presume that it is supposed to guarantee to find (in particular) a 25-sphere configuration in 4D if one exists. I can't really see how it can do that, while still being approximate. Do you think it can?

I don't think that one of these epsilon-algorithms could be guaranteed to find a 25-sphere configuration if one exists, because if the algorithm required exact placement, it wouldn't be found, no matter how small epsilon was. But I can imagine an algorithm of this sort with the following property: If a 25-sphere configuration exists, then the algorithm will find 25-sphere configurations for spheres of size 1 - epsilon that all touch a central sphere of size 1, no matter how small epsilon is. Finding such configurations for small epsilon would lend credence to the theory that there is a 25-sphere configuration with spheres of size 1, and examination of the configuration of spheres of size epsilon may help guess the 25-spheres-of-size-1 configuration. Conversely, if there is no 25-sphere configuration, then compactness shows that there is no 25-sphere configuation for spheres of size 1 - epsilon for sufficiently small epsilon, so a program of this sort could provide a proof if epsilon is chosen correctly. Andy Latto andy.latto@pobox.com