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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