Erik D. Demaine, Martin L. Demaine, Anna Lubiw, Arlo Shallit, Jonah L. Shallit

professors and students
MIT and University of Waterloo
Boston MA, USA and Waterloo ON, Canada

This work is a collaboration involving Erik D. Demaine (MIT), Martin Demaine (MIT), Anna Lubiw (Waterloo), Arlo Shallit, and Jonah Shallit. The goal was to do math with family members: Erik is Martin's son, and Arlo and Jonah are Anna's sons.

We were exploring what 3-dimensional shapes can be made from a planar polygon and one zipper. Or, to put it another way, what 3-dimensional shapes can be unzipped to lie flat in the plane without overlap.

Felt models with real zippers are a satisfyingly tangible way to appreciate the problem.

zipper unfolding of an octahedron
5" x 5" x 5" (3d); 12" x 16" (2d)
felt with zipper, plus colour photocopy
2010

An octahedron can be unzipped to a planar polygon.

In fact, all the Platonic and Archimedean solids have zipper unfoldings (http://erikdemaine.org/zippers/).

Can every convex polyhedron be unzipped to a planar shape? If the zipper is restricted to travel along the edges of the polyhedron, then a Hamiltonian path is necessary (though not sufficient) and the answer is "no". The general question is open.

zipper unfolding of a double pyramid
5" x 5" x 5" (3d); 9" x 15" (2d)
felt with zipper, plus colour photocopy
2010

On the outside, this model is an octahedron. However, it has an extra square face on the inside, so it is really two pyramids glued together on their bases. This two-celled polyhedral complex has a zipper unfolding.