Fred Lunnon wrote:
I put in some time [too much, probably] on rational simplices over the holiday[...]
Just the opposite here: I was hoping to rewrite my (slow Mma-based) search in gp/pari, but never found the time. (Surely there is more than one ambiguous hexad!) The real problem with not being a math professor any more: no grad students to run off and do this kind of thing for you.
5 vertices in 3 dimensions [...] there is also the single ambiguous pair
[7, 6, 2, 6, 2, 3, 5, 4, 4, 4], [7, 6, 2, 6, 2, 3, 4, 5, 4, 4].
Very nice to see! As with the only 4-points-in-2d example we have, this is a proper simplex plus one point with ambiguous placement. This seems like the odds-on bet for where to look for ambiguous cases, especially ones with more than two configurations. I wonder if there's a good way to build a search that looks for that situation specifically? In particular, given a proper simplex (with, say, integer edge lengths), is there an easy (eg finite) way to tell whether there is a set of (let's say rational) distances from its vertices resulting in ambiguous placement of the last point?
6 vertices in 3 dimensions --- none for edges up to 10. Although there are numerous pairs which may be glued together at a common tetrahedron, either the single new edge created is irrational, or else one of the component tetrahedra turns out to be planar. Almost certainly a considerably more extensive search is required to find such an object, if it exists.
My intuition is that this, like the abiguous-planar case, would be better suited to a search targeted at finding those pairs to glue, rather than looking for an example with small edge lengths. --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.