It's fairly clear that this problem will fall soon; somebody will give an exact construction of an example. I'm pretty sure I've seen origami models with zigzag pleats that could probably be extended and bent into a torus with little effort. That having been said, it seems to me that this problem would be an excellent exercise in genetic algorithms. Specify a "polyhedron" as a topological complex, with points at given coordinates, edges between given pairs of points, faces among given cycles of edges. Genetic operators would shift points slightly, add or delete edges, and mutate the incidence relations. The fitness function would penalize nonplanar or self-intersecting faces, and faces that interpenetrated each other, as well as violations of the necessary incidence relations; it would reward small numbers of vertices and of course the correct genus. This is only a sketch; it would be a nice seminar term project to fill in the details and see if the approach found any interesting small examples. On Sun, Aug 9, 2009 at 1:39 PM, James Propp <jpropp@cs.uml.edu> wrote:
Oops: make that 4 vertices, 12 edges, and 8 faces (check: 4-12+8=0).
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