I think 8 faces is possible: Connect two opposite edges of a regular tetrahedron ABCD with a segment I, 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. —Dan
On Oct 13, 2015, at 8:02 AM, James Propp <jamespropp@gmail.com> wrote:
2) Call a face of a polyhedron "supporting" if it's part of a support plane of the polyhedron (that is, the polyhedron lies in one of the closed half-spaces determined by the plane) and "unsupporting" otherwise, and call a polyhedron "totally unsupported" if it has no supporting faces. What is the smallest number of faces a totally unsupported polyhedron can have? My record is 12 (see below).