Dürer’s Unfolding Problem

Image: Wikimedia Commons

Can every convex polyhedron be “unfolded” into a shape that doesn’t overlap itself, like this dodecahedron?

Surprisingly, no one knows.

The Renaissance artist Albrecht Dürer first wondered about this in the early 1500s, and in the ensuing five centuries no one has found a polyhedron that won’t permit it. But one may yet exist.