Erik Demaine's Folding and Unfolding:

Folding Polygons into Polytopes

Erik Demaine, Martin Demaine, Anna Lubiw, and Joseph O'Rourke

The basic problem we have been studying is in what ways a polygonal piece of paper can have its boundary glued to itself and then be folded to form a convex polyhedron. This line of research was initiated in the paper ``When Does a Polygon Fold to a Polytope?'' by Anna Lubiw and Joseph O'Rourke. The problem is also intrinsically related to Aleksandrov's theorem, which gives a simple characterization of what gluings lead to convex polyhedra, and says that a gluing can lead to only one convex polyhedron.

For more information on our work, see

Our video on the subject, Metamorphosis of the Cube, showing the 5 edge-to-edge gluings of the Latin cross, and other unusual unfoldings of the cube.
An example, the 85 foldings of the Latin cross.
Our recent technical report documenting some of our work, Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes (July 2001). A shorter version of this paper, re-organized to focus on the combinatorial results and not include the examples and counterexamples, appears in the Japan Conference on Discrete and Computational Geometry (November 2000).

More information to come soon.

Last updated November 28, 2010 by Erik Demaine. — Accessibility