for example the one shown in Figure \[steiner\].

The result was fascinating to watch: with $3$ or $4$ pegs, the optimum tree usually *is* found.  However, by no means is it always found, especially with more pegs.  Soap-bubble partisans might write this off as experimental error, caused (for example) by inaccuracy in placing the pegs, or by the interference of my hands.  However, I also sometimes found triangular bubbles of three Steiner vertices—which is much harder to explain, since such a structure could never occur in a Steiner tree.  In general, the results were highly nondeterministic; I could obtain entirely different trees by dunking the same configuration more than once.  Sometimes I even obtained a tree that did not connect all the pegs.

Another unexpected phenomenon was that sometimes the bubbles would start in a suboptimal configuration, then slowly relax toward a better one.  Even with $4$ or $5$ pegs, this process could take around ten seconds, and it is natural to predict that with more pegs it would take longer.  In short, then, I found no reason to doubt the party line, that soap bubbles do not solve $\mathsf{NP}$-complete problems in polynomial time by magic.[^5]

There are other proposed 