There was a brief period in my life when I was obsessed with the universal constant Pi just as much as Max Cohen (Sean Gullette) was obsessed with finding the number that was the answer to the universe in Pi. That period was during my last undergraduate semester, where we devoted our efforts to estimating Pi to as many digits as possible for our Numerical Analysis class, which culminated in a conceptual painting about the irrational.
Pi isn't about the mathematical constant 3.1415926..., representing, among many other things, the ratio of the circumference of a circle to its diametre. The movie is about the deranged and beautiful quest of one person in search of the truth, the answer to the universe. The plot is a common one in science fiction: a phenotypic "aberration" in the brain causes the protagonist to develop special abilities that makes him sought after and feared. In this particular cases, Max acquires a deep grasp of number theory. With his assumptions, that mathematics is the universal language, that number theory can represent everything in nature, and that there is a pattern in everything that occurs in this universe, he sets about trying to find it in the stock market.
After him are people who are interested in his stock market analyses for monetary purposes, and more strangely, a group of Rabbis who are convinced the same pattern of numbers is the key to their salvation. However, Max is the only one who can understand the semantics of the 216 digit number that is key to the universal lock, a plot device that I thought was truly brilliant. In the end, Max succeeds on his quest, but what he understands is never revealed to us. What he sees however is catalyst enough for him to inflict a lobotomy upon himself.
Today, many scientists (including myself) are on the same quest that Max is, and most don't need to be as obsessed to find what they're looking for. As demonstrated several times over the course of humanity's existence, there is indeed an explanation for a lot of things we see in nature. That is, at least a conceptual level, there do exist patterns that can be written out as mathematical equations. This is particularly true given the discoveries this centuries involving relativity, quantum mechanics, and the subsequent biological and computing revolutions. But the search for the Grand Unified Theory continues.
Even though the movie isn't directly about Archimedes' constant, Pi is indeed an excellent solution to Max's problem. That constant is ubiquitous in our world today, popping up every so often, in a seemingly independent manner. Besides its strong presence in geometry, Pi appears in various equations throughout mathematics (especially certain infinite series) and even in places where you wouldn't necessarily expect it to at first intuition. For example, the Buffon needle problem: what is the probability that a needle of length 1, thrown at random on a plane divided by parallel lines 1 unit apart, will land in such a manner that it crosses a line (2/Pi)? What is the probability that two integers chosen at random have a common factor exceeding one (6/Pi^2)?
Unlike many independent films, Pi actually has a fairly coherent, albeit obscure, plot. The cinematography is excellent as well. Filmed completely in black and white, with obscure and surreal settings, the movie tries very hard to brings the audience to the edge of the abyss that is Max's mind. The acting by Sean Gullette is highly convincing. The electronic music (courtesy of Clint Mansell of Pop Will Eat Itself) is top-notch. The movie is definitely worth the approximately Pi * e dollars I paid for it and I highly recommend checking it out at a local independent theatre near you.