between them are dipped into soapy water, then the soap bubbles will rapidly form a Steiner tree connecting the pegs, this being the minimum-energy configuration.

\[ptb\]

[steiner.eps]{}

It was only a matter of time before *someone* put the pieces together.  Last summer Bringsjord and Taylor [@bt] posted a paper entitled P=NP to the arXiv.  This paper argues that, since (1) finding a Steiner tree is $\mathsf{NP}$-hard, (2) soap bubbles find a Steiner tree in polynomial time, (3) soap bubbles are classical objects, and (4) classical physics can be simulated by a Turing machine with polynomial slowdown, it follows that $\mathsf{P}=\mathsf{NP}$.

My immediate reaction was that the paper was a parody.  However, a visit to Bringsjord’s home page[^3]  suggested that it was not.  Impelled, perhaps, by the same sort of curiosity that causes people to watch reality TV shows, I checked the discussion of this paper on the comp.theory newsgroup to see if anyone recognized the obvious error.  And indeed, several posters pointed out that, although soap bubbles might reach a minimum-energy configuration with a small number of pegs, there is no magical reason why this should be true in general.  By analogy, a rock in a mountain crevice could reach 