I was amused by the work of Scott Aaronson on soap bubbles and Steiner trees. Given $n$ points $p_1$, $p_2$, ..., $p_n$ in $\mathbb{R}^2$, the Steiner tree through these points is the connected planar graph of shortest length containing these points. Computing the topology of the Steiner tree, given the coordinates of the points, is NP-hard.
On the other hand, if two parallel glass plates are separated by rods in positions $p_1$, $p_2$, ..., $p_n$ and dipped into a bucket of soapy water, the shape of the resulting soap film is a local minimum for this problem. Which lead to the question: How close is the local minimum, formed by whatever complicated PDE governs soap film, to the NP-hard global minimum?
Aaronson decided to try the experiment, and reported on his results in Section 3 of "NP-complete problems and physical reality". The answer was not very close: The stable shape of the soap film was often not even a tree!
