Bill Gosper <rwmgosper@yahoo.com> wrote: Last night Alan Schoen sent me a solution so much more efficient than my tricuspid that it worked in all four orientations! And here I thought line thickness and the weirdness of the cavity would prevent even copying, let alone solution, given just that lo-res drawing . I'm tempted to ask Alan to sit on his arrangement while I try to "tricuspidize" it. --rwg Partial success: A huge effort produced a tricuspid packing, but with undesirable radii. The good news is that there are three degrees of freedom in the equations. Small matter of additional effort. The main difficulty is with arnoldp(x,y,r), a 3367 term polynomial which vanishes if a circle at (x,y) of radius r is tangent to the ovoid. Unfortunately, not iff. The only way I could eliminate extraneous parameters and reduce arnoldp to three variables was to numerically approximate the coefficients. Then the elimination process (aided by Mathematica, since my copy of Macsyma was compiled for a 286 in NT) introduced huge spurious factors. The false solutions are raising hell with Newton's method, several of whose 36 equations are arnoldps. It's taking forever, and screwing up. The 3367 floating point coefficients preclude symbolically extracting the bogus factors. But DUH! With a fish dinner and a shot of chai, I just realized an obvious way for Arnold to visit Jenny Craig. If any funsters chirp up with the answer, I'll kick myself for not requesting help earlier. If not, stay tuned for more arnage. --rwg PS, Cool mental factorization of 3367 (one third, two thirds): 10101/3 = 111111/3/11 = 111/3 1001/11 = 37 7 13. --------------------------------- Looking for last minute shopping deals? Find them fast with Yahoo! Search.