It is interesting that some of the most-amazing among the listed miracles involve only powers of 2 in denominators: 5*acos(-7/8)+2*acos(-61/64) =1080.0 4*acos(-57/64)+5*acos(-31/32) =1440.0 1*acos(57/64)+5*acos(-3/4) = 720.0 1*acos(61/64)+5*acos(-1/4) = 540.0 4*acos(-3/4)+1*acos(-31/32) = 720.0 2*acos(-3/8)+1*acos(-23/32) = 360.0 2*acos(5/8)+1*acos(7/32) = 180.0 3*acos(-17/32)+4*acos(-11/16) = 900.0 1*acos(-7/32)+2*acos(-5/8) = 360.0 1*acos(17/32)+4*acos(1/4) = 360.0 1*acos(23/32)+2*acos(3/8) = 180.0 1*acos(31/32)+4*acos(3/4) = 180.0 3*acos(1/4)+1*acos(-11/16) = 360.0 3*acos(-1/8)+2*acos(-9/16) = 540.0 2*acos(-11/16)+3*acos(-7/8) = 720.0 1*acos(-9/16)+3*acos(-3/4) = 540.0 2*acos(9/16)+3*acos(1/8) = 360.0 1*acos(11/16)+3*acos(-1/4) = 360.0 2*acos(-1/4)+1*acos(-7/8) = 360.0 2*acos(3/4)+1*acos(-1/8) = 180.0 1*acos(1/8)+2*acos(-3/4) = 360.0 1*acos(7/8)+2*acos(1/4) = 180.0 There also are examples involving only powers of 3: 2*acos(-4/9)+1*acos(-49/81) = 360.0 2*acos(-2/9)+1*acos(-73/81) = 360.0 2*acos(5/9)+1*acos(31/81) = 180.0 2*acos(8/9)+1*acos(-47/81) = 180.0 4*acos(-22/27)+3*acos(-79/81) =1080.0 1*acos(-31/81)+2*acos(-5/9) = 360.0 1*acos(-17/81)+4*acos(-1/3) = 540.0 1*acos(17/81)+2*acos(-7/9) = 360.0 1*acos(47/81)+2*acos(-8/9) = 360.0 1*acos(49/81)+2*acos(4/9) = 180.0 1*acos(73/81)+2*acos(2/9) = 180.0 1*acos(79/81)+4*acos(-2/3) = 540.0 3*acos(-2/3)+1*acos(-22/27) = 540.0 3*acos(1/3)+1*acos(-23/27) = 360.0 3*acos(-7/9)+2*acos(-23/27) = 720.0 3*acos(1/9)+2*acos(-22/27) = 540.0 1*acos(22/27)+3*acos(2/3) = 180.0 1*acos(23/27)+3*acos(-1/3) = 360.0 2*acos(-1/3)+1*acos(-7/9) = 360.0 2*acos(2/3)+1*acos(1/9) = 180.0 1*acos(-1/9)+2*acos(-2/3) = 360.0 1*acos(7/9)+2*acos(1/3) = 180.0 what is going on here? I will try to rerun the program out to 128 and post the results later. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)