At 05:59 PM 9/28/03 -0400, John Conway wrote:
After some more reflection, I've come to the conclusion that Allan's "Tightness Conjecture" probably WILL fail in 3 dimensions as well as in 4, and that Jud can probably redeem himself (so to speak) by establishing this.
Ah. I misread you (JHC) in your last message, or you thought better of it.
Namely, among the many packing configurations that Sloane et al have produced with the "Gosset" program, there are probably several records that aren't (currently) achieved by any "tight" configuration. Now if Jud were to run his program for spherical caps of the appropriate radius (minus, for safety, a very small epsilon), it should prove this.
... The expectation being that JM's program would fail to find any (tight) configurations, even though non-tight solutions for those cases are known to exist. Yes, that would be very cool.
It would also be of interest to run it for the case of 4-dimensional unit spheres, to see just how far short of 24 it falls. Let me just conjecture here that it won't even find 23.
I'm sure the program would also find a few unforeseen things that would amply justify the trouble of writing it. How about it, Jud?
Well, it's certainly a fairly tough programming challenge. I'm not volunteering to write it! Can JHC say a few words more about the "Gosset program"? I have other ideas for computer experiments in this area, and would like to know if they've been tried already. -A