On Sun, 28 Sep 2003, Allan C. Wechsler wrote:
JM proposed, if I read him right, to prune this space by considering only configurations [that can] be built by adding each new sphere in the 'socket' formed by three previous spheres.
JHC expresses doubt that this program could work, because of the possibility of losing the baby with the bathwater. It might be possible for 25 non-overlapping 4-spheres to kiss a central one, but only in configurations that are _not_ in the restricted class JM proposes.
Not just doubt, but moral certainty. The problem is not that there might be 25-sphere configurations, but no tight ones, but that there IS a 24-sphere configuration that is presumably the only one, and it ISN'T tight.
Obviously, the JM program depends on an unstated conjecture, one that JHC believes to be false. If forced to place bets at this point, I'd wager that JHC is right ... unless JHC offered me attractive odds. Because I'm not yet morally certain he's right. The conjecture, which I will call the Tightening Conjecture, is: if there exists ANY configuration of n non-overlapping radius-R k-spheres kissing a given unit k-sphere, then there exists a TIGHT such configuration.
JHC puts forward what I believe is intended to be a counterexample to this conjecture, but I claim it isn't really a counterexample.
I was aware of this - it can be deformed into an antiprism. The 4-dimensional example doesn't suffer from such a defect. The example's virtues were that it was 3-dimensional, and sufficient to disprove Jud's method (by which I mean only "show that it's not obviously correct").
The Tightening Conjecture may still be false, but JHC's cubical example does not kill it outright. My guess is that JHC has a stronger counterexample in his arsenal, that _will_ disprove the TC; my only real point in this message is that the offered example doesn't do the job.
If Musin's proof is correct (as I expect it will be), and proves the 24-sphere configuration unique (as I expect it to), then that does disprove the TC. This reminds me that I had intended to ask, but forgot, a question that I can now rephrase as "is there any radius r for which there is a packing of N circular caps of radius r on the unit sphere, but no tight one?" I don't expect there will be, but unlike the two expectations above (and like the TC), this expectation has no real basis. John Conway