In practice, it just seems to happen that if you do not fix a triangle, the only solutions you find have pairs of equal length. Perhaps light might be cast on the subject via an exhaustive analysis of the formal solutions of --- say --- the (essentially) 29_C_2 = 406 possible subsets of equations for a triple planar hexad. Uuurghhh! WFL On 11/30/06, Michael Kleber <michael.kleber@gmail.com> wrote:
It only just occurred to me that the geometric constructions we've seen so far for doubly planar charts (no rationality requirement) all build families that preserve a single triangle, and provide alternate placements for the fourth point.
Fred's various examples -- integers with the collinearity blemish and generic quartics -- both do likewise.
Is this necessary? That is, given the six distances, can you at least identify three of them that must form a triangle, or might there be two planar charts with no triangles in common? (It's possible that someone has posted an example I've missed which already settles this question; I haven't tried to do an exhaustive literature search :-).