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Problem 9: EdgeUnfolding Convex Polyhedra
 Statement
 Can every convex polyhedron be cut along its edges and unfolded
flat to a single, nonoverlapping, simple polygon?
 Origin
 First stated in [She75], but in
spirit at least goes back to Albrecht Dürer [Dür25].
 Status/Conjectures
 Open.
It seems to be a widespead hunch that the answer is YES.
 Partial and Related Results
 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].
 Related Open Problems
 Problem 42: VertexUnfolding Polyhedra.
Problem 43: General Unfolding of Nonconvex Polyhedra.
Problem 64: EdgeUnfolding Polyhedra Built from Cubes.
 Appearances
 [She75,O'R00,MO01]
 Categories
 folding and unfolding; polyhedra
 Entry Revision History
 J. O'Rourke, 2 Aug. 2001.
 BDE+03

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):5162, 2003.
 Dür25

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'.
 MO01

J. S. B. Mitchell and Joseph O'Rourke.
Computational geometry column 42.
Internat. J. Comput. Geom. Appl., 11(5):573582, 2001.
Also in SIGACT News 32(3):6372 (2001), Issue 120.
 O'R00

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 258266. SpringerVerlag, 2000.
 She75

Geoffrey C. Shephard.
Convex polytopes with convex nets.
Math. Proc. Camb. Phil. Soc., 78:389403, 1975.
The Open Problems Project  December 08, 2012