Not so the sextic -4536767107584 + 2746424942208 x - 464573095920 x^2 + 14000936832 x^3 + 744041160 x^4 + 710712 x^5 + 169 x^6 for the 12 circle optimal packing http://www2.stetson.edu/~efriedma/cirRcir/12.gif, "6 dimes & 3 quarters around 3 nickels", which is what I Moebius transformed to make Twubblesome Twelve, naively complaining to this list at the time that the equation(s) seemed unduly complicated. So now, with the miracle of sextic solving, we have the solution in radicals, except I can't simplify it enough to fit on one page. --rwg On Thu, Oct 27, 2011 at 3:46 AM, Bill Gosper <billgosper@gmail.com> wrote:
maximizing sum(radii), http://www2.stetson.edu/~efriedma/cirRcir/5.gif, namely Root[-87131 + 32779 #1 + 18567 #1^2 + 10931 #1^3 - 14350 #1^4 + 2538 #1^5 + 162 #1^6 &, 2], is unsolvable in radicals, according to my (Ssexy) solver. --rwg