Spelling: Sonsky --> Somsky. Despite Somsky's 6-planet animations, I do not believe they necessarily represent exact solutions and in fact I already emailed Somsky a calculation about irrational angles indicating one of his animations was in fact NOT an exact solution, but rather contained some phase mismatches. My latest theorem I posted, note, only proves existence with approximate phase angle matches, which in general are not exact. The claim Rokicki has that if all gears have integer radii, then exact meshability happens, I do not anymore believe. (I'd "proved" it, but Somsky showed my proof involved a sign error.) As Somsky himself pointed out, this is a necessary but not sufficient condition; the question of whether suitable "phase angles" exist then must be answered. But Rokicki's argument about planets with one pair of them being "Somsky flexible" indeed makes sense to me, and indeed what it tells me is, in my proof about approximate and exact meshability, whenever you have an unexpected extra degree of motional freedom -- such as Somsky's sun+antisun+2 planets flexible configuration where the sun, amazingly, can move its rotation axis -- you get an extra amount of exact meshability. Specifically, my latest proof I posted allegedly indicates that if the planar bipartite graph G has 1 interior face, then we can get exact meshability in the Euclidean plane, and if it has 2 interior faces, then we can get exact meshability in nonEuclidean planes. But if we now add Somsky-esque flexibility to the mix, that should enable boosting "1 face" and "2 faces" to "2" and "3" respectively, whenever we can embed a suitable subgraph of G in an "unexpectedly flexible" way. The sun+antisun+N planets configuration corresponds to a graph G which is bipartite with 2 pink vertices (sun & antisun) and N blue ones (planets) and N-1 interior faces (each a quadrilateral) and 1 exterior face (also a quad). So my general purpose theorem applied to this planets situation, predicts exactly meshing solutions should happen in Euclidean plane when N=2, and when the first 2 planets happen to obey Somsky's flexibility condition, then when there are N=3 planets. Further, for bevel gears (spherical geometry) my general purpose theorem predicts exact meshing should happen when N=3, and if the first 2 planets happen to yield a flexible situation (which I'm not sure is possible) then when N=4. Rokicki seemed to think he could add a pair of planets at once, but that is not clear to me, maybe his flexing-based argument only allows him to add 1 planet. (Rokicki curx paragraph which may be wrong: "At this point you just insert the larger planets of the two Somsky gears; they are guaranteed to mesh with the sun and the outer planet, so you are done." Really?)