The New Yorker recently ran a feature on how a couple of mathematicians helped The Met photograph a part of The Hunt of the Unicorn tapestries. That same week, they ran from their extensive archives a 1992 profile of the same mathematicians, brothers David and Gregory Chudnovsky. The Chudnovskys were then engaged in calculating as many digits of pi as they could using a homemade supercomputer housed in their Manhattan apartment. There's some speculation that director Darren Aronfsky based his 1998 film, Pi, on the Chudnovskys and after reading the above article, there's little doubt that's exactly what he did:
They wonder whether the digits contain a hidden rule, an as yet unseen architecture, close to the mind of God. A subtle and fantastic order may appear in the digits of pi way out there somewhere; no one knows. No one has ever proved, for example, that pi does not turn into nothing but nines and zeros, spattered to infinity in some peculiar arrangement. If we were to explore the digits of pi far enough, they might resolve into a breathtaking numerical pattern, as knotty as "The Book of Kells," and it might mean something. It might be a small but interesting message from God, hidden in the crypt of the circle, awaiting notice by a mathematician.
The Chudnovsky article also reminds me of Contact by Carl Sagan in which pi is prominently featured as well.
According to Wolfram Research's Mathworld, the current world record for the calculation of digits in pi is 1241100000000 digits, held by Japanese computer scientists Kanada, Ushio and Kuroda. Kanada is named in the article as the Chudnovskys main competitor at the time.
(Oh, and as for patterns hidden in pi, we've already found one. It's called the circle. Just because humans discovered circles first and pi later shouldn't mean that the latter is derived from the former.)