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.