Yes, I had already decided myself Rokicki was right, etc, in correspondence to him, Somsky, Deventer; which I'd also mentioned in a post to math-fun. But Julian Ziegler-Hunts may also be confused, in the sense there are TWO kinds of motion possible in Somsky's configuration, namely (a) turning the gears, (b) moving the sun-gear's position. My error was only considering motions of type (b); JZH may be making the opposite error. Type-b motion inherently has bounded travel. Type-a allows unlimited turning. The animations at https://www.youtube.com/watch?v=ApD2pSTWVjk and http://gosper.org/blockysomsky2.gif make the latter especially clear, also showing other people were also working on this. This also confirms my general-N planets claim in two particular cases. The main results I am aware of at present at (1) Somsky theorem about movable sun (2) the analogous (?) less interesting theorem that if you make a quadrilateral ABCD out of 4 meshed spur gears, then it cannot flex -- except in the degenerate case where A & C are racks, then the motion is arguably a bit more interesting than usual; and it perhaps is interesting that both this & Somsky's case involved a two-angles-are-zero degenerate quadrilateral; (3) my general theorem that any planar bipartite graph is attainable as a gear-meshing graph for gears in the plane that do not intersect, and can be turned in sync thru infinite angle. And this attainability in a countably infinite number of inequivalent ways for each such graph. Not yet answered: (4) I presume/hope, but have not investigated/checked, that Somsky further showed his theorem was best possible, i.e. the sun can never move unless his condition is obeyed? I.e. in a quadrilateral of 4 meshed gears, 3 being spur, the 4th being an "inner ring" complemented-spur, the only way the quadrilateral can flex, is if Somsky condition obeyed? So it is "if and only if"? (5) One could also consider bevel gears, which naturally live not on a plane, but on the surface of a sphere; their axes all pass thru the sphere center. Then re-pose all the same questions in that scenario. [True masochists could also consider the hyperbolic plane as the remaining nonEuclidean geometry.] In particular, re my old proof ("theorem 2") about unsymmetric planetary gear systems... I think it should be possible to construct functioning unsymmetric planetary bevel gears on a sphere, I now outline how to try. I am not sure if this now will be possible for any number N of planets; on the sphere there might be a finite maximum possible N. The "stereographic projection" maps sets of circles on the plane to sets of circles drawn on the sphere, and it maps rational coordinates to rational coordinates. https://en.wikipedia.org/wiki/Stereographic_projection http://www.isallaboutmath.com/proof.aspx (annoying long video of ster-proj proof, couldn't bear to watch it all the way thru; but lovely graphics & audio) However, it usually maps rational circle-radii to irrational circle radii (radii measured angularly). That seems like it is usually going to kill you for gear purposes. However, then you realize that what matters for gears is not the angular radius R of the circle drawn on the unit sphere, but rather its CIRCUMFERENCE, which equals 2*pi*sin(R). So what matters is the sine of the angular radius. Since sin(2*arctan(x)) = 2*x/(1+x*x) is a rational function, it is quite possible to map circles with rational radii in the plane, to circles on the unit sphere whose angular-radii have rational sine. In fact this ALWAYS happens if the plane-circle is centered at the origin which is the south pole of the sphere. Then sin((a-b)/2) = sqrt( (1-cos(a-b))/2 ) = sqrt( (1+sin(a)*sin(b)-cos(a)*cos(b))/2 ) indicates that in order to make it happen for off-center circles, you need to be somewhat lucky, the luck is similar to the luck needed in choosing a random right triangle with integer legs and hoping it has integer hypotenuse. So anyhow, I suspect this amount of luck should be attainable after computer search, at least for some small number of planets. (6) My (3) totally settles the question of which "topologies" (or "meshing combinatorial structures," whatever you want to call it) are attainable for functioning coplanar gears. However if we allow gears of different thicknesses and in offset parallel planes, then nonplanar graphs also become possible and I do not have a characterization of which graphs then are attainable. This question likely is too difficult to solve completely, but it is probably feasible to say some interesting stuff that partially solves it. In particular, Kuratowski's theorem tells us that in some (admittedly not very relevant) sense the "only" nonplanar graphs are K5 and K33, and only the latter is bipartite. K23 is attainable as a planetary gear system with 3 planets, 1 sun, and 1 outer ring. Indeed we can even achieve "K33 with a single edge missing." Is it possible to achieve K33 as circle tangency relations among 6 distinct circles in the plane? Yes... but the trouble is that one of them is going to have to be tangent on BOTH its inside AND outside, which is a pretty strange kind of gear, so I do not think you can do this mechanically. If we ask for K33 where each edge is allowed to be a gear CHAIN not just two gears directly meshed, then it is no problem to add the missing edge of K33 in the form of a chain with 2 intermediate gears. It similarly is no problem to do K5 where each K5 edge is done as a chain with 1 or 3 intermediate gears. So I think therefore that any graph can be instantiated using gears provided each graph edge is implemented as a long-enough gear chain, not a direct meshing.