(To Dan Asimov: On the sphere S2, these are bevel gears, and are highly realistic, useful, applicable, etc. On the hyperbolic nonEuclidean plane H2, though, I'm not seeing any realistic use.) Inversive transformations and the stereographic projection all preserve the gear-touch/not combinatorial relations, and preserve rationality of euclidean-radius ratios. Somewhat more interesting is: COSINE RATIONALITY PRESERVATION LEMMA: Doing an inversive map to a set of gear-circles in the plane (to get another such set) can preserve the rationality of cosines. More precisely, if gear A has unit-radius and center (0,0) contacts gears B and C, and the angle on A between the two touch points, has rational cosine, and the AB contact point is (0,1), and the center of inversion has rational coordinates then the cosine will still be rational after the inversive mapping It is conceivable this lemma will enable proving something amazing about Somsky gear existence.... but also conceivable it will lead nowhere. The following problem seems related: POINT SETS WITH INTEGER DISTANCES. It has been conjectured that if N points in the plane exist, with all interpoint distances integer, then either (a) N is upper bounded by some finite constant C (b) the points all lie on the union of a line and a circle As far as I know this conjecture has never been proven, although occasionally somebody raises the record C by one more point. I'm not sure what is the latest greatest record C, but it is at least 6 and I'm pretty sure nobody has managed to reach 10. Update: I think the latest record is C=7, see this 2008 paper: http://arxiv.org/abs/0804.1303v1 Obviously, any two gears in a meshed set have integer center-separations if those two gears mesh. Also by the "Kepler law" Somsky planet centers all lie on an ellipse hence NOT a line U circle if enough of them. So this while not exactly the same problem, it strongly suggests (at least, for those who believe that Conjecture) that is it NOT going to be possible to find an N-planet Somsky planet system in the Euclidean plane with unboundedly great N. There will be some finite upper bound on N. The lack of progress during the last 70 years on the integer-distance problem makes one worry, though, that proving this "Somsky nonexistence conjecture" might also be infeasibly hard. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)