Re: [math-fun] A use for Jud's method
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
On Sun, 28 Sep 2003, Allan C. Wechsler wrote:
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.
Maybe a bit of both. At the moment I dreamed up the "cube" example, I thought of it as rigid (or, to be more precise, didn't think at all!), but by the time I was finishing the letter, I'd realised it could be deformed into the antiprism (with change of radius). So there might have been some too-strong language in the earlier part of that letter.
... 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.
"Gosset" takes as input the name of some kind of "space", and an integer N, and tries to fit N points into the space so as to maximize their minimal distance. It does this by throwing them in at random to begin with, and then cleverly jiggling them around to increase the distance. The examples for the sphere that I mentioned were obtained by running it thousands of times for each N, and selecting the best configuration found. JHC
I'm getting very intrigued to know what number Jud's method would produce in the 4-dimensional case. The point there is not that the method MIGHT not produce a 25-sphere example that MIGHT exist (but almost certainly DOESN'T), but that it WON'T produce the 24-sphere example that definitely DOES exist (and is almost certainly UNIQUE). This is because that configuration has the shape of the regular polytope {3,4,3}, from which you can see that its cells are octahedra rather than the tetrahedra that Jud's method requires. Maybe the method won't even find so many as 20 spheres? John Conway
participants (2)
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Allan C. Wechsler -
John Conway