Delete "however" in previous post! Incidentally, note that the polytore is a counterexample to a claim on Weisstein's page that << Furthermore, for a polyhedron with no coplanar faces, at least one edge cut must be made from each vertex or else the polyhedron will not flatten. >> Possibly that page was intended to refer to _convex_ polyhedra alone? [ If anyone out there knows his email address, please forward this to him! ] WFL On 6/25/17, Fred Lunnon <fred.lunnon@gmail.com> wrote:
See however http://mathworld.wolfram.com/Unfolding.html etc. ---
<< Shephard's conjecture states that every convex polyhedron admits a self-unoverlapping unfolding (Shephard 1975). This question is still unsettled (Malkevitch), though most mathematicians believe that the answer is yes.
It is known that Shephard's conjecture is false for non-convex 3-dimensional polyhedra (Bern et al. 1999, Malkevitch). >>
WFL
On 6/25/17, Adam P. Goucher <apgoucher@gmx.com> wrote:
<< 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 ...
If you swap the existential quantifier with a universal one, you get an unsolved problem:
"Does every convex polyhedron have at least one net (unfolding without overlaps)?"
...so I imagine that, for this to be unsolved, there must be convex polyhedra for which there is at least one unfolding with overlaps.
Best wishes,
Adam P. Goucher
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