At 04:05 PM 9/28/03 -0400, John Conway wrote:

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 [of the negative]. 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.

THIS is the crucial piece of information I was missing when I wrote my previous message. Glad I didn't bet. I didn't realize that the known 24-sphere configuration was (a) believed to be unique, and (b) was not tight.

[...] 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.

Indeed.

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.

In other words, your intuition is that the TC is true for the case of 3 dimensions; is that right? -A