Re: [math-fun] Kissing number (again).
At 08:11 AM 9/26/2003, John Conway wrote:
In fact, they were quite valid, except that I think I mistakenly described the number of cases for each sphere (after the first few) as "a 3-dimensional continuum", when it's really only 2-dimensional. Jud's method is indeed fallacious.
In 3 dimensions, it would probably get the correct kissing number of 12, and so maybe leave the fallacy undetected, but almost certainly the fallacy would be made manifest in the 4-dimensional case by the fact that it would "prove" the kissing number to be < 24.
My 4-D intuition is certainly no good, so I was wondering if what seems would work in 3D would carry over to 4D (about whether or not it is sufficiently general for a ball to touch the maximum number of other balls). But let me ask this about the "continuum", and I'll use the 3D analogy. Suppose you've got a few balls touching the central one and you're finding the places where you can add another one. The simplest approach would be to find all of the places where it touches the central ball and two others, but that isn't sufficiently general. Consider the new ball touching the center ball and is in contact with ball A. Swing it one way (staying in contact with ball A and the central ball) until it touches ball B. Swing it the other way until it touches ball C. Now would it be the case that only certain areas in that swing need to be considered? That is, suppose that the new ball swings through x degrees going from B to C. Would it be that, say, any position from 0 to 0.1x is essentially the same because it makes no essential difference in how the rest of the balls are placed, and 0.1x to 0.36x are essentially the same, etc? If so, that would be how it can be reduced to a finite number of cases. Like I said, I have no 4D intuition, so maybe higher than 3D is to unconstrained for that to work.
On Fri, 26 Sep 2003, Jud McCranie wrote:
At 08:11 AM 9/26/2003, John Conway wrote:
In fact, they were quite valid, except that I think I mistakenly described the number of cases for each sphere (after the first few) as "a 3-dimensional continuum", when it's really only 2-dimensional. Jud's method is indeed fallacious.
In 3 dimensions, it would probably get the correct kissing number of 12, and so maybe leave the fallacy undetected, but almost certainly the fallacy would be made manifest in the 4-dimensional case by the fact that it would "prove" the kissing number to be < 24.
My 4-D intuition is certainly no good, so I was wondering if what seems would work in 3D would carry over to 4D (about whether or not it is sufficiently general for a ball to touch the maximum number of other balls). But let me ask this about the "continuum", and I'll use the 3D analogy.
I should have made it clear that your argument is invalid in 3 dimensions just as much as it is in 4 - the difference is that only that I believe it would get the correct answer in 3 dimensions, whereas it fairly certainly won't in 4. To see why, let's use the usual reduction of the problem to packing spherical caps into a sphere, and let's change the size of these caps to that at which there's room for just 8 mutually tangent ones, centered at the vertices of a cube. Then after starting with two touching ones of these, your method would place a third so as to touch both of them, which would prevent it from finding the correct configuration. So it's not just that your intuition has failed in 4D but works in 3D - it's really a wrong method that might accidentally find the right answer in 3D, but wouldn't serve of a proof of it.
Suppose you've got a few balls touching the central one and you're finding the places where you can add another one. The simplest approach would be to find all of the places where it touches the central ball and two others, but that isn't sufficiently general. Consider the new ball touching the center ball and is in contact with ball A. Swing it one way (staying in contact with ball A and the central ball) until it touches ball B. Swing it the other way until it touches ball C. Now would it be the case that only certain areas in that swing need to be considered? That is, suppose that the new ball swings through x degrees going from B to C. Would it be that, say, any position from 0 to 0.1x is essentially the same because it makes no essential difference in how the rest of the balls are placed, and 0.1x to 0.36x are essentially the same, etc? If so, that would be how it can be reduced to a finite number of cases.
Like I said, I have no 4D intuition, so maybe higher than 3D is to unconstrained for that to work.
...and as I said, that's not what's wrong or right with it. The reason I believe it will actually give the wrong answer in the 4-dimensional case is that in the conjecturally unique optimal configuration there happen to be three mutually tangent balls, but not 4. So on the supposition that the known 24-ball configuration is indeed uniquely optimal, I KNOW you won't be able justify the method's assumption that the fourth ball can be chosen to touch the first three. In fact it's obvious that the method will find a 12-ball solution in the 3D case, because I can actually see that how to successively place the 12 balls so that each after the second touches two previous ones. But that merely means that it will produce a fallacious proof of a true result, and is not a great recommendation! John Conway
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John Conway -
Jud McCranie