Before Bill Thurston steps in and cleans up here, I think I'll just put in my two-penn'orth ... A rather elegant class of degenerate solutions --- almost certainly more disreputable than you had in mind --- are provided by "kaleidocycles", or rotating rings of tetrahedra. See Marcus Engel's elegant "AniKa" animations at http://www.kaleidocycles.de/anim.shtml I'd guess that you found an example with 12 vertices, 24 edges, 12 faces which are all self-intersecting quadrilaterals --- though I must admit to not having actually computed any coordinates! Fred Lunnon On 8/7/09, James Propp <jpropp@cs.uml.edu> wrote:
Here's a puzzle for which I found a nice but degenerate solution: Find a toroidal polyhedron of genus 1 whose corners are all "flat" in the sense that the angles of the faces meeting at each corner add up to 360 degrees.
(I agree that there a number of inequivalent ways to interpret the problem, but I don't know how to clarify the statement of the problem without giving too big a hint!)
I'm curious whether equally nice but less degenerate examples are known.
Jim
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