rwg>How about your semisecret fourteen disk solution,
which has no symmetry at all? (http://milou.msc.cornell.edu/images/>>>> seems to be [intermittently] down.)>>>> --rwg
Whoa, what about twelve? http://gosper.org/HTMLFiles/12disks.gif
I thought this was the "six dimes and three quarters around three nickels" I Mobius-transformed to make Twubblesome Twelve, but it seems to have been Mobius transformed already! It appears to retain bilateral symmetry, and the transformation should be described by a single real parameter. If I can reconstruct my "six dimes" formulae, it should become a Calculus I problem with an exact algebraic solution. --rwg
For (unbeknownst to me) a stupid technical reason, this discussion shrank off-list to just Veit, David Cantrell, and me. And then took some surprising turns. Using commercial software, David has improved upon Veit's 10, (11, slightly?), 12, 13, and 14 packings, which I thought had been proved optimal! See http://www2.stetson.edu/~efriedma/cirRcir/ In "14", the extremely close pair that Veit reported is no longer the closest. The new closest differ substantially more, but are probably (uncomfortably) swappable, even in aluminum, and might need to be artificially equalized to make a decent physical puzzle. And my proposed Calculus I problem for "12" is obviated--David finds the optimal Möbius transformation is the identity. He sent the polynomial for the exact Sum(radii), which factors into nasty cubics. David's packings go all the way to "32", with "31" perhaps the most amazing, for lack of the D(ihedral)_5 symmetry that "32" almost has. Talk about a sucker bet-- who would believe the "31" concentric shell packing was suboptimal? <Insert Baskin-Robbins joke here.> So the "14" puzzle is somewhat ruined, and the next totally asymmetric packing is "25" (! Almost inconceivable.) Are we out of metagrobological luck? Probably not--many of David's bilaterally symmetric packings have about the same number of different sized pieces as Veit's "14"! Veit adds: VE>I should explain why my symmetries were off. I never doubted the 12 packing had D_3 symmetry. My code requires a target value for the radius sum (it's a constraint solver). My procedure was to converge on the optimum by bracketing the optimal radius sum on a sequence of shrinking intervals. Clearly I terminated my sequence too early for the 12 packing in that the imperfect convergence didn't escape your notice!
I don't have a good excuse for missing the obvious 10 packing!
Veit --rwg