Unlocking Puzzling Polygons - geometric research - Brief Article

Science News, Sept 23, 2000 by Ivars Peterson

"This allowed us to apply some beautiful theorems in rigidity theory," Demaine notes.

A proof that flat polygonal chains can't lock followed from that insight. Along the way, Demaine, Rote, and Connelly also established that any open chain can always be straightened.

The big surprise is not the proof itself, O'Rourke comments, but the conceptual breakthrough that the opening move in any successful uncrinkling process has to be one in which each joint moves apart or stays the same distance away from every other joint.

Last summer, Demaine, Connelly, and O'Rourke added another element to the original argument. They showed that the area inside an uncrinkling polygon must increase. "This seems almost obvious," Connelly notes, "but the proof that we have is not completely trivial."

Now that the two-dimensional case is solved, Demaine is tangling with other fierce geometric beasts. An origami enthusiast, he's tamed the hyperbolic paraboloid. Demaine developed instructions for folding and gluing this classic saddle shape into complex paper hats and starbursts.