On Fri, Oct 10, 2008 at 2:08 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Let B = B_0 be a unit ball in 3-space that's tangent to the z-axis.
Let K > 0 be an unknown but fixed constant.
Now inductively, for all n > 0, place B_(n+1) so that it's tangent to both B_n and the z-axis, so that its center's z-coordinate exceeds that of B_n by K.
QUESTION: --------- Find K_min := the least K such that all nonempty intersections between two B_n's are tangencies.
--Dan
Fun! When K=0, B_2 is the same as B_0, so K>0. When K > 1/2, B_1 can't be placed tangent to B_0. When K = 1/2, there's only one way to place them and they're tangent, so 0 < K <= 1/2. For K=1/4 and B_0 centered on (1/2, 0, 0) we get x = +/-1/2, y=0 for odd indices and x=0, y=+/-1/2 for even indices greater than 0. There'll be a smaller K where you get a spiral of balls, but I'm not sure how to work out what it is. -- Mike Stay - metaweta@gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com