However, the pink vertices happen to be the sun and outer-ring gear (which I'll call the "antisun") in planetary systems; the planets then are the blue vertices. So long as the planets viewed from the sun are spaced at rational angles apart (measured in degrees) and ditto for the antisun, we are ok. Indeed it suffices if each sun-viewed angle plus its corresponding antisun-viewed angle, is a rational.
--indeed in planetary system, if the sun-antisun angle viewed from each planet is rational (in degree measure), that suffices. And necessary. So this is leading us to the question of understanding triangles whose sidelengths and whose angles in degrees, all simultaneously are rational numbers. Google tells me this question has been looked at already by "Lubin": http://math.stackexchange.com/questions/389139/can-all-possible-angles-on-a-... which claims that the only possible such triangles have angles that are multiples of 30 degrees. This would seem to prove 12 is the most possible planets (or an upper bound on same) for an unsymmetrical planetary... at least if this condition were effectively necessary not merely sufficient. http://rangevoting.org/WarrenSmithPages/homepage/works.html paper #72 also considers somewhat related question plus cites some papers Lubin could/should have cited.