[Veit's packings:> http://milou.msc.cornell.edu/images/ ] [David's packings: http://www2.stetson.edu/~efriedma/cirRcir/ ]
[...] rwg>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 did not have the exact polynomials for "22", which also factor (over {sqrt 7,(-1)^(1/7)}) down to nasty cubics, tasting which Corey's denester suffers bovine parturition. rwg>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?
Given the consistent superiority of symmetric packings in the smaller
cases, I couldn't resist checking David's numbers by exactly solving the D_5 "31" case, which proved challenging. It's easy to write six trig and pythagorean equations for the six unknown radii, but not so easy (at least for me) to clear the radicals, after which Reduce has been working for days to "univariablize" (triangularize, then back substitute) the six polynomials. But there's a workaround: Guess the radius equations (they're all degree 20, the one for the center disk being reciprocal) with PSLQ and then vet them with RootReduce, which is even now busily checking that sum(radii) is also degree 20. Stay tuned for algebraic details. --rwg