It finally occurred to me to compute the dimension of the solution set. Consider an arbitrary "octocog" as the union of a pair of disc subsets comprising given radii p,p',p",r and q,q',q",s respectively. Modulo isometry, each subset has freedom 2 (eg. discs p', p" move freely relative to disc p while touching disc r ); fitting them together imposes none for q to touch p',p" ; 2 for q' to touch p,p" , 2 for q" . Hence suitable disc configurations have dimension 2*2 - 2*2 = 0 --- so finitely (though not in this case denumerably) many for given radii. Now a gear train must satisfy 6 additional belt constraints (around discs s, q', q", p etc). Only 5 of these are linearly independent (any belt is effectively the path sum of the remaining 5 ); there might conceivably be further nonlinear dependencies between the constraints. But even a single independent constraint is sufficient to reduce the freedom of the solution set to less than zero --- hence any octocog is inevitably going to be special or degenerate in some way, eg. the symmetric touching specimen discussed earlier. Similar considerations apply to analogous "(2 n)-cogs" for n = 3,4,... , irrespective of imposed symmetry: the solution set has negative dimension, so any solution must degenerate in some way. Seems the snark really was a boojum after all ... Fred Lunnon On 10/23/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
I hunted the "octocog" with 6-fold symmetry: like the snark, it remains elusive.
Such a gear train would comprise a sun gear centred at the origin, of radius s ; which meshes with three counter-rotating inner planets of radius q ; which each mesh also with two of three outer planets of radius p ; which all mesh with an external ring of radius r .
The constraint that corresponding discs should touch reduces to 4*X^2 - 3*Y^2 = W^2 (*) where X == p-q , Y == s-q , Z == r-p , W = 2 Z - Y . The meshing constraints involve belts around discs s,q,p,q and r,p,q,p resp.
Essentially the only solutions found for X < 1024 are scalar dilations of [X, Y, Z, W] = 12 [1, 1, 1, 1] corresponding to concentric increments of the degenerate [r, s, p, q] = [18, 6, 6, -6] --- six equal planets touching both a sun and a ring of triple radius. Any increment of this has overlapping discs.
The diophantine tangency equation (*) has some independent interest. Apparently a complete set of primitive, non-negative solutions is generated (after reducing by GCD and eliminating repetitions) by the set of three parametric forms [X, Y, Z, W] , [X, Z, Y, X+Y-W] , [X, |Y-Z|, Y, X+Y] (**) where X = u^2 + 3*v^2 , Y = 4*u*v , W = 2*|u^2 - 3*v^2| , Z = (Y+W)/2 .
The first few primitive solutions are 1, 0, 1, 2 1, 1, 1, 1 7, 3, 8, 13 7, 5, 8, 11 7, 8, 5, 2 13, 7, 15, 23 13, 8, 15, 22 13, 15, 8, 1 19, 5, 21, 37 19, 16, 21, 26 19, 21, 16, 11 31, 11, 35, 59 31, 24, 35, 46 31, 35, 24, 13
This is a special case of the natural extension of Pythagorean triples to solutions of (say) a X^2 = b Y^2 + c Z^2 ; however I could not locate any relevant discussion --- references, anyone?
Fred Lunnon
On 10/19/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
This problem impinged during my customary brainstorming session in search of convincing reasons not to get out of bed. I have not examined it, so have no idea whether it is well-known, easy, hard or impossible.
Deventer's "Gear Shift" puzzle http://www.jaapsch.net/puzzles/gearshift.htm is topologically equivalent to an octahedron with a gear on each face, meshing with three gears adjacent at edges.
Is it possible to morph this concept into its planar map: a flat train comprising one ring enclosing two 3-planet tiers around one sun?
Furthermore, do such trains exist with 6-fold, 2-fold, and no symmetry?
Fred Lunnon