I assumed Jim's example to be at the very least (equivalent to) a simplicial complex immersible in Euclidean 3-space, and that he was asking for an embedding ... maybe he can enlighten us? Allan's origami suggestion seems as good a definition of the non-degeneracy constraint as one might require. WFL On 8/10/09, Dan Asimov <dasimov@earthlink.net> wrote:

This polyhedron discussion has been severely hampered by the absence of a definition of polyhedron. (There are many different definitions.)

E.g., imagine a square with edges drawn from the center to its 4 corners and also to its 4 side-midpoints. Now identify opposite sides of the square in the usual way to get a torus. You end up with 8 faces, 12 edges, and 4 vertices.

--Dan

WFL wrote:

<< On 8/9/09, James Propp <jpropp@cs.uml.edu> wrote:

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Oops: make that 4 vertices, 12 edges, and 8 faces (check: 4-12+8=0).

I don't think this can be correct. The only polyhedron with 4 vertices is a tetrahedron (4-6+4 = 2). WFL

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