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Can every convex polyhedron be cut along its edges and unfolded flat to a single, nonoverlapping, simple polygon?
First stated in [She75], but in spirit at least goes back to Albrecht Dürer [Dür25].
Open. It seems to be a widespead hunch that the answer is yes.
The answer is known to be no for nonconvex polyhedra even with triangular faces [BDE+03], but has been long open for convex polyhedra [She75], [O'R00].
Problem 42: Vertex-Unfolding Polyhedra.
Problem 43: General Unfolding of Nonconvex Polyhedra.
Problem 64: Edge-Unfolding Polyhedra Built from Cubes.
[She75], [O'R00], [MO01]
folding and unfolding; polyhedra
J. O’Rourke, 2 Aug. 2001.
Marshall Bern, Erik D. Demaine, David Eppstein, Eric Kuo, Andrea Mantler, and Jack Snoeyink. Ununfoldable polyhedra with convex faces. Comput. Geom. Theory Appl., 24(2):51–62, 2003.
Albrecht Dürer. The painter’s manual: A manual of measurement of lines, areas, and solids by means of compass and ruler assembled by Albrecht Dürer for the use of all lovers of art with appropriate illustrations arranged to be printed in the year MDXXV. New York: Abaris Books, 1977, 1525. English translation by Walter L. Strauss of ‘Unterweysung der Messung mit dem Zirkel un Richtscheyt in Linien Ebnen uhnd Gantzen Corporen’.
J. S. B. Mitchell and Joseph O’Rourke. Computational geometry column 42. Internat. J. Comput. Geom. Appl., 11(5):573–582, 2001. Also in SIGACT News 32(3):63-72 (2001), Issue 120.
Joseph O’Rourke. Folding and unfolding in computational geometry. In Proc. 1998 Japan Conf. Discrete Comput. Geom., volume 1763 of Lecture Notes Comput. Sci., pages 258–266. Springer-Verlag, 2000.
Geoffrey C. Shephard. Convex polytopes with convex nets. Math. Proc. Camb. Phil. Soc., 78:389–403, 1975.