Linkages have a long history ranging back to the 18th century in the quest for mechanical conversion between circular motion and linear motion, as needed in a steam engine. In 1877, Kempe wrote an entire book of such mechanisms for "drawing a straight line". (In mathematical circles, Kempe is famous for an attempted proof of the Four-Color Theorem, whose main ideas persist in the current, correct proofs.) Kempe designed many linkages which, after solidification by modern mathematicians Kapovich, Millson, and Thurston, establish an impressively strong result: there is a linkage that signs your name by simply turning a crank.

Over the years mathematicians, and more recently computer scientists, have revealed a deep mathematical and computational structure in linkages, and how they can fold from one configuration to another. In 1936, Erdös posed one of the first such problems (now solved): does repeatedly flipping a pocket of the convex hull convexify a polygon after a finite number of flips? This problem by itself has an intriguingly long and active history; most recently, in 2006, we discovered that the main solution to this problem, from 1939, is in fact wrong.

This talk will describe the surge of results about linkage folding over the past several years, in particular relating to the two problems described above. These results also have intriguing applications to robotics, graphics, nanomanufacture, and protein folding.