π

1. Mathematics is the language of nature.
2. Everything around us can be represented and understood through numbers.
3. If you graph these numbers, patterns emerge. Therefore: There are patterns everywhere in nature.

—Maximillian Cohen

A brilliant mathematician and his home-built supercomputer Euclid search for the underlying patterns of the universe.

Max Cohen is a reclusive, paranoid, migraine-afflicted genius driven by the unshakable belief that the universe can be explained entirely through math. Chased by agents of a Wall Street firm interested in profiting from his work as well as a sect of Hasidic Jews who want him to help find patterns in the Torah, Max stumbles upon a 216-digit number that seems to hold some universal secret. As he continues his research, his headaches worsen and he begins experiencing vivid hallucinations. Though he pushes on, determined to puzzle out the number's meaning, his mind might not be ready to comprehend it.

Shot in minimalist black and white, the film is mostly atmosphere, with relatively little dialogue. At its core, it's a Mind Screw with a Gainax Ending, but an intriguing one indeed. The whole experience is often compared - favourably - to Eraserhead.

π is the directorial debut of Darren Aronofsky, who also wrote the film. It was made for a mere $60,000 and won the 1998 Sundance Film Festival award for Best Director. In his first film-score work, the movie's soundtrack was assembled by Clint Mansell of "Lux Aeterna" fame, who contributed the frenetic drum-and-bass theme.

Tropes used in Π include:

All There Is to Know About "The Crying Game": The infamous "drill" scene.
Animal Motifs: Max finds ants as well as goo infesting his apartment after he loads it into his computer Euclid. A possible interpretation is that the number, which is the blueprint for all creation, causes Euclid to begin creating simple organic matter. The goo and ants also fit into the themes of madness and decay that increasingly plague Max.
Book Ends: "When I was a little kid my mother told me not to stare into the sun. So once when I was six, I did" is said at the beginning and end of the film.
Combat Pragmatist: Max prefers to run when Marcy Dawson's thugs but at one point, they chase him into a grocery store where he grabs a can of soup and uses it to mass effect.
Cyberpunk
Deliberately Monochrome: And filmed in ultra-high-contrast for added creepiness.
Earn Your Happy Ending: The ending leaves you with more questions than answers, but at least Max is happy, one way or another.
Facecam: One of the more infamous uses of the so-called SnorriCam. The DVD contains some footage of a test run with Max in a convenience store.
Fade to White: Most notable in the climax.
Formulaic Magic: Max's dogmatic treatment of math as "the language of nature" seems to push past "math is capable of explaining everything in existence" into "math determines our existence." The effects of the number point to the latter.
Gainax Ending: Max is shown to trepan himself at the end of the movie, voiding himself of his mathematical genius. Whether this is literal or symbolic is left up to the viewer.
Hacker Cave: Euclid takes up the majority of Max's apartment. His landlady doesn't much like it.
Hates Being Touched: Max.
Hikikomori: Max is a shut-in.
Important Haircut: Max shaves his head as things start going downhill. Of course, there's a reason he does it.
Instant AI, Just Add Water: What happens after Max connects the mysterious Ming-Mecca chip to Euclid: the extravagantly-cheap, lovingly-customized mainframe (threatening to overrun every square centimeter of Max's tiny fortress of an apartment). That is, if one can believe Sol--outwardly rationalistic, yet painfully aware of forbidden mysticism--when the former professor explains how the 216-digit number makes machines "aware of their silicon nature."
Intelligence Equals Isolation: Max's quest for the universal number has left him utterly disconnected from the rest of humanity, to the extent that he even fears leaving his apartment if he could run into someone.
Logic Bomb: The universal number for both computers and people.
Lucky Charms Title
Mad Mathematician: Max.
Madness Mantra: Max "restates his assumptions" several times throughout the film, which includes his various theories as well as the story of how he started getting migraines after looking into the sun too long.
Madness Montage: The "hip-hop montages" of surreal imagery.
Mind Screw: The A.V. Club called it "like Eraserhead re-envisioned by Cyberpunk author William Gibson."
Mouthful of Pi: Not strictly pi, but Max can do rather complicated arithmetic in his head.
No Budget
Non-Indicative Name: Neither the actual number π, nor what it represents (the ratio of the circumference of a circle to its diameter), is important to the plot of this film at all.
The Professor: Sol, Max's mentor and one of the few people Max actually talks to.
Rage Against the Reflection
Religion Is Magic: Played straight and subverted in equal measure. The Kabbalistic Jews know about the universal number, and it is related to the Torah, but they are unable to discover it themselves. In the end, they are no closer to it than a stockbroking firm.
Scary Black Man: Gender Flipped with Marcy Dawson.
These Are Things Man Was Not Meant to Know: The 216-digit number repeatedly crashes Euclid and drives Max insane. It's also implied that Sol's second stroke came about because he started researching the number again. It's repeatedly linked with the sun, which fried Max's optic nerves when he gazed at it and killed Icarus when he flew too close to it.
Video Full of Film Clips: A music video for the soundtrack's "πr^2" is included on the DVD, with shots from the film interspersed with random color footage of ants (a major recurring motif).
Writers Cannot Do Math:
- Several mathematical facts get mixed up (for example, the golden ratio is usually phi, not theta), but it doesn't detract too much from the plot.
- Max guesses that the Jews have already "intoned" every possible name of God, which the rabbi does not deny. This would be completely impossible if each name were a 216 digit number. However, earlier Lenny shows that words in Hebrew have numeric values based on the sum of the values of their letters, so the Jews might actually be looking for a word that adds up to 216. Barring some unmentioned grammatical limitations, it would probably still be impossible to intone every single possible permutation.