The recent Carpenter's Rule Theorem (2000) says that a polygonal linkage,
made up of rigid bars connected into a chain, can always be folded into any
possible configuration, while avoiding crossings between the bars and while
preserving the bar lengths. We develop a new approach to folding such
linkages based on following the gradient flow of a suitable energy function.
This approach has several advantages over previous approaches in terms of
mathematics, algorithms, and practice. On the mathematical side, we prove
existence of a $C^\infty$ motion. On the algorithmic side, we give the first
finite algorithm to construct an explicit piecewise-linear motion. On the
practical side, our algorithm is straightforward to implement, possibly even
physically without a computer.