Please ignore previous (brainstorm) caveat: I was right first time, so was TR. However if instead grandad is simply scaled up by 4, the sunset becomes 10. There is now room along the axis to insert two more planets, yielding the 8-planet train [28, -12; 12, 8, 4, 3; 4, 8, 12, 13] . Scaling this up by 2 again then decreasing by 3 yields [53, -27; 21, 13, 5, 3; 5, 13, 21, 23] , the train reported by WRS on July 13th. I conjecture that (assuming nonzero sunset) every 8-planet train is thus constructable; and further that there are no trains with more than 8 planets. [ By the way, using "offset" to denote displacement of sun from ring centre turns out badly, for reasons best postponed. Similarly --- as previously hinted --- I should like from now on to change the sign of my sun radius: please be patient and indulge me! ] Fred Lunnon On 8/5/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
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.