At 01:24 PM 9/26/2003, Dan Hoey wrote:
Consider the new ball touching the center ball and is in contact with ball A.
You can't assume that. It might be in touch with just one of the previously-placed balls, and you can't even guarantee which one.
I meant that it is in contact with some ball, call that ball "A" (not that it was in contact with a particular ball). (Similarly B and C are not particular balls.)
I looked at trying to modify this for four dimensions, and I find that you can prove that every ball must touch at least two others. Same in K dimensions: every ball must touch at least two others. You might have all N balls in a loop, so you have to guess each ball but the first, second, and last with K-1 degrees of freedom.
Too much freedom.
You can't do better with "geometric intuition". John was right, of course.
I think you're both right, and thanks to everyone for the insight.