Problem 9: Edge-Unfolding 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: Vertex-Unfolding Polyhedra.
Problem 43: General Unfolding of Nonconvex Polyhedra.
Problem 64: Edge-Unfolding Polyhedra Built from Cubes.

Appearances

[She75,O'R00,MO01]

Categories

folding and unfolding; polyhedra

Entry Revision History

J. O'Rourke, 2 Aug. 2001.

Bibliography

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):51-62, 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):573-582, 2001.
Also in SIGACT News 32(3):63-72 (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 258-266. Springer-Verlag, 2000.

She75

Geoffrey C. Shephard.
Convex polytopes with convex nets.
Math. Proc. Camb. Phil. Soc., 78:389-403, 1975.