Caveat: the accompanying gloss, while correct in a general sense, overlooks the mildly embarrassing fact that radius-3 planets of the original actually coincide on the axis! The radius-15 partner therefore lies opposite along the axis. The Somsky pair subsequently attached have (maximal) equal radius-9 . This illustrates a special case of a construction to which I want return shortly --- I had intended to delay until the discussion of coaxial / concentric families had finished, but events seem to be overtaking me! WFL On 8/5/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Bang on cue!
This baby reduces to a "grandfather" rational train as follows: (39, 21, 3, 3) ; Replacing one planet by its Somsky opposite, then completing a third planet on the same side as the other two -> [39, 21; 15, 9, 3] ; Scaling down by factor 3 -> [13, 7; 5, 3, 1] ; Increasing concentrically -> [14, 6; 6, 4, 2] ; Scaling down by factor 2 -> [7, 3; 3, 2, 1] ; Final offset = 5/2 .
I conjecture that any rational train is reducible to the grandfather in a similar fashion.
Fred Lunnon
On 8/5/15, Tom Rokicki <rokicki@gmail.com> wrote:
Here's one solution based on one of Warren's miracles.
Plot (39, 21, 3, 3) (ring, sun, planet, planet).
This gives an offset of exactly 15.
This is due to the odd equation
2 * acos(-11/16) - 3 acos(7/8) = pi
Integral offsets such as these are rare except in certain degenerate cases.