Excellent question, which had not occurred to me. Can the argument be refined to establish upper or lower bounds, or better still exact values, for the (maximum) distance travelled? [Memo to self: if in a hurry, do not accept lifts in vehicles designed by anyone called Deventer, Somsky, etc.] WFL On 7/9/15, Warren D Smith <warren.wds@gmail.com> wrote:
Oskar Deventer in his video remarks he can make Somsky gears with 1 sun and 4 planets, and also says Somsky showed one can make 1 sun and 6 planets.
I point out that it is possible to make Somsky gears with 1 sun and 2N planets for ANY value of N>0, and in a countably infinite number of ways for each N. Begin with a symmetric centered sun and all planets of equal size. This is trivial to do for any value of N. Now perform a "geometric inversion" transformation http://mathworld.wolfram.com/Inversion.html https://en.wikipedia.org/wiki/Inversive_geometry to the whole configuration, causing the gear radii to become all unequal and making it unsymmetric. However if all the radii originally were rational and the coordinates of the gear centers ditto, and ditto for all the parameters of the inversive transformation (which we of course can easily assure; there is a dense infinite set of rational-coordinate points on the standard unit circle corresponding to pythagorean triples) then all that will still be true post-transformation. And note inversions map circles to circles. And then by scale-up by the LCM of the denominators we can make all radii be integer, hence result will work as Somsky gears for a limited but nonzero travel distance. QED.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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