<< Is this minimal number a function of the number of faces, edges and vertices of the polyhedron? Is this minimal number unique? >> It's unclear over what set these questions are intended to minimise! One possible interpretation is answered negatively by the following. << Does any unfolding of the same polyhedron have the same number of edge cuts? >> No. In an August 2009 math-fun thread instigated by Jim Propp's request for a `polyhedral origami torus' with angular defect zero at every vertex, I proposed a family of `polytores' having 20 faces (4 trapezia + 16 triangles), 12 vertices (8 5-valent + 4 6-valent), 32 edges. Developing the planar net may yield either 14 = 2*7 or 16 = 2*8 free edges, depending on which vertices are interior. See https://www.dropbox.com/s/t8iqaeoe5e86ld1/solitore3.pdf https://www.dropbox.com/s/42grmh6o3re4ulf/flattore3.pdf << is there a polyhedron where some unfolding has more edge cuts, but shorter total edge cut length, than some other unfolding? >> Yes. A tall, narrow polytore net may exchange 4 short trapezium edges for 3 long triangle edges. << Is there a convex polyhedron for which some unfolding exhibits overlapping faces in the plane? >> Intuitively, `unfolding' can only increase the distance between (given points on) any two faces. However, it's not at the moment obvious to me exactly why this should be a consequence of convexity ... Fred Lunnon On 6/25/17, David Wilson <davidwwilson@comcast.net> wrote:
1. For a given polyhedron, what is the minimal number edges that need to be cut to unfold it into a connected planar surface? For example, 3 edges are necessary for a tetrahedron, I think 7 for a cube.
2. Is this minimal number a function of the number of faces, edges and vertices of the polyhedron?
3. Is this minimal number unique? Does any unfolding of the same polyhedron have the same number of edge cuts?
4. If (3) is false, is there a polyhedron where some unfolding has more edge cuts, but shorter total edge cut length, than some other unfolding?
5. Is there a convex polyhedron for which some unfolding exhibits overlapping faces in the plane? If so, what is the smallest number of faces on such a polyhedron?
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