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