There's a bug in the JZH animation: the sun gear is slightly up-scaled. An animation of Deventer's 3-planet "Looney gears" (sic) lacking the extra radius-13 planet is at https://www.youtube.com/watch?v=ApD2pSTWVjk The Somsky proof at 3:35 in Deventer's video https://www.youtube.com/watch?v=M_BUn4TDns8 establishes that the gear teeth actually mesh together correctly. In an elementary 3-planet mechanism, this meshing imposes two additional (continuous) constraints on the mechanism, leaving only a _single_ degree of freedom modulo isometry for given planet radii. It follows that if a frame of reference is chosen in which (say) centres of outer ring and one planet are fixed in the plane, then all centres are necessarily fixed; the remaining freedom is expressed by the rotation angle of the ring. Fred Lunnon On 7/12/15, Bill Gosper <billgosper@gmail.com> wrote:
On Sat, Jul 11, 2015 at 11:59 PM, Julian Ziegler Hunts <julianj.zh@gmail.com
wrote:
Rokicki's right; Smith's misreading the proof (more specifically, I think he's noticing that two angles are the same, but thinking that those two angles change when the mechanism is rotating, which they don't—they're related to the relative positions of the gears).
The hard parts in animating gears are (i) the fact that gears don't actually move as if they're non-slipping adjacent circles and (ii) deciding what the teeth should look like, paying attention to the fact that, because (i) is most easily solved by ignoring it, you have to put effort into making it look realistic (no intersections, no gaps).
I mostly ignored these problems in creating the attached gif, which shows a complete cycle. It's not very smooth, but it gets the point across and it's already 1.6MB.
Julian
gosper.org/blockysomsky2.gif Superb. Thank you! Considering the spectrum of possible working configurations, it looks like the revolution (not rotation) period of the sun gear goes to infinity as its radius of gyration goes to zero. Another pathway to van Deventer's 11 million to 1 reducer? --Bill _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun