What constraints determine whether a vector represents some realisable train? If sunset h = 0 the train is conventional with sun and ring concentric, and if h < 0 the axis may be reversed; so assume h > 0 . All radii will be nonzero integers, positive except for s . The sun and ring will not overlap; however, restricting h a priori to prevent planets overlapping does not currently appear feasible. Experimentally it appears that the overall structure of realisable X is quite simple. Each of the following cases is a subset of the case preceding; cases with unequal numbers of planets on each side can always be completed via Somsky's lemma. Proofs are mundane and available but omitted. ** Case 2 planets: Somsky's lemma shows X = [ r, s; p ; u ] with r + s = p + u and (say) p >= u is realisable for the continous interval of sunsets max( 0, 2 p - r - s, r + s - 2 u ) < h < r + s . ** Case 4 planets: X = [ r, s; p, q; u, v ] with r + s = p + u = q + v and p > q is realisable for some finite nonempty set of h , in the smaller interval deducible from previous case. ** Case 6 planets: X = [ r, s; p, t, u; u, t, p ] with r + s = p + u = 2 t and p > t > u is realisable for some finite nonempty set of h , in the previous interval. ** Case 8 planets: X = a Z + b I , where I = [ 1, 1; 1, 1, 1, 1; 1, 1, 1, 1 ] , Z = [ 1, -1; 1/5, 0, -1/5 -1/4; -1/5, 0, 1/5, 1/4 ] , h = 1/2 is realisable for integers a, b satisfying previous restrictions. A computer survey of trains with r <= 50 supports the ** Conjecture: Every realisable eccentric train falls under some case above! In particular, it would follow that there are no trains with 10 planets. Fred Lunnon On 8/6/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
Incidentally, under a fairly mundane extension of mechanical realisability, the sun is allowed partially to overlap the ring, creating a new crescent-shaped region in which planets rotate counter to those remaining in the old crescent. The new planets require negative radii for previous formulae to work smoothly.
WFL
On 8/6/15, Fred Lunnon <fred.lunnon@gmail.com> wrote:
On 8/6/15, William R Somsky <wrsomsky@gmail.com> wrote:
Umm.... Ah!
You're setting X_2 = -(sun radius) so that I is all ones, rather than [1, -1; 1, ..., 1], yes?
There's a little more to it than that. My radius sign indicates direction of rotation (rather than of teeth, as one might naïvely expect). With this convention, concentric increment corresponds to the "offset" operation in spherical geometry, in which each oriented sphere (including points and hyperplanes) changes in radius by a constant amount.
WFL