On Mon, 29 Sep 2003, Richard Schroeppel wrote:
One curiosity: There are two instances where K and K+1 points have the same angle, indicating that removing a point from the K+1 packing doesn't make room to expand the remaining K points. These are 5,6 and 11,12. I didn't see any more cases up to 130 points, the table limit.
In the case 5,6 the equality is provable - it follows from a theorem of Rankin. I once wrote out most of a proof for the 11,12 case, and think I could complete it for sufficient payment. I don't believe there will be any more such N,N+1 cases for the 3D problem. Here's a nice theorem of mine along these lines if 6 non-overlapping pennies are in some circle, then they can continuously be moved around inside that circle so as to make room for a seventh penny. The 12,13 3D analog of this fails, but I think the 24,25 version might be valid in 4D.
The higher dmensions had some longer runs, where K...K+5 all had the same minimal angle.
The same theorem of Rankin will account for some of these. Regards, JHC