[Delayed in vain anticipation of Julian's blessing.] Fred, stop polishing those damned specs! The bug isn't the sizes, it's the physically impossible square teeth, as Julian admitted. Another reason to hate orthodontists. It's ironic that mathematically correct teeth sell the mathematically incorrect Looney gears. --rwg JZH animation: gosper.org/blockysomsky2.gif Somsky's 5 planets(!): gosper.org/somsky5.gif -------- Original Message -------- Subject: Re: [math-fun] math-fun Somsky gear controversy Date: 2015-07-12 04:02 From: Fred Lunnon <fred.lunnon@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Reply-To: math-fun <math-fun@mailman.xmission.com> There's a bug in the JZH animation: the sun gear is slightly up-scaled. An animation of Deventer's 3-planet "Looney gears" (sic) lacking the extra radius-13 planet is at https://www.youtube.com/watch?v=ApD2pSTWVjk The Somsky proof at 3:35 in Deventer's video https://www.youtube.com/watch?v=M_BUn4TDns8 establishes that the gear teeth actually mesh together correctly. In an elementary 3-planet mechanism, this meshing imposes two additional (continuous) constraints on the mechanism, leaving only a _single_ degree of freedom modulo isometry for given planet radii. It follows that if a frame of reference is chosen in which (say) centres of outer ring and one planet are fixed in the plane, then all centres are necessarily fixed; the remaining freedom is expressed by the rotation angle of the ring. Fred Lunnon