On Mon, 29 Sep 2003, Dan Hoey wrote:

John Conway <conway@math.princeton.edu> 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?

No, of course not. Why would you presume I attempt such a guarantee, when it is patently impossible?

Because I thought you must, if you're to be able to establish that 24 is an upper bound. But I'm still in quarantine, so maybe I'm confused.

I claim that if there are no (25,4,1+2epsilon) configurations, the algorithm would say so. This is important, because if there are no (25,4,1) configurations, there must be an epsilon for which there are no (25,4,1+2epsilon) configurations.

I also claim that every (25,4,1) configuration is represented in the output by an approximation that describes the coordinates of the centers to within epsilon. Each of these can be converted to a (25,4,1-2epsilon) configuration.

Aha. In other words, if there is a 25-sphere solution, the method won't itself decide either that or its negative, but would produce something that we'd probably be able to recognize as interesting. JHC