At the conclusion (oh yeah?) of an cringingly embarrassing catalogue of blunders in numerical math., prog. logic, and software & mech. eng. I have succeeded in confirming Tom Rockiki's exact Spirograph solution; and must now grovel for having previously cast doubt on it. WFL On 7/23/15, Tom Rokicki <rokicki@gmail.com> wrote:
Center of outer gear is at (0,0).
Information below is (x, y, r, theta).
Gear 0: 0 -15.2455256731227 24 0
Gear 1: -36.7084008061191 -54.8497339123657 30 1.29754164761468
Gear 2: 50.6617258475364 -39.1074102204309 32 -1.82681164698356
Gear 3: 36.7084008061191 39.6042082392429 42 0.961032493387983
Gear 4: -50.6617258475364 23.8618845473082 40 -1.58285857438843
And here it is in spirograph. http://tomas.rokicki.com/somsky-spiro.jpg (This may not be exactly the solution I calculated but it does work in plastic).