Another 12-face solution is a throwing star made by joining four very acute triangular pyramids so that their equilateral sides form a tetrahedron. But that'll be moot if Dan's 8-face solution checks out. Dan wrote: I think 8 faces is possible:
Connect two opposite edges of a regular tetrahedron ABCD with a segment I,
I assume that Dan has in mind that I joins the midpoints of those two edges (not that it matters). and let the 1/3 and 2/3 points of I be 2 new vertices E and F.
Now throw away all faces and the 2 edges that I connects, and draw the 8 new edges between each of E and F with A, B, C, D. The total of 12 edges define 8 triangular faces.
I am having trouble picturing this. Can Dan or someone else suggest an alternative way to see this polyhedron (say by gluing pieces together)? Jim