The modern history of Pi is no less exciting than the ancient history.
In 1999 Yasumasa Kanada calculated 206,158,430,000 decimal digits of Pi, setting a world record, only to top it himself three years later.
In 2002 his team at the University of Tokyo calculated 1,241,100,000,000 digits. This calculation took more than 602 hours on a Hitachi SR8000 computer and required more than 1 terabyte of memory (duh!). Those of you who do not have access to SR8000 may like to know that the 1,241,100,000,000th digit after the decimal point is 5. The results were used to calculate the frequency of all ten digits in Pi and they all are quite close to 10%. Of course, this is not enough to prove that the Pi is normal (that is that every sequence of numbers of certain length has the same probability of appearing in Pi).
But another approach is likely to prove more fruitful.
In 1995 David Bailey, Peter Borwein, and Simon Plouffe made an exciting discovery. They found a simple formula for calculation of Pi that incidentally allowed independent calculation of any single hexadecimal digit of Pi. This formula (named BBP formula after the scientists who discovered it) is provided below:
Pi = SUMk=0 to infinity 16-k [ 4/(8k+1) - 2/(8k+4) - 1/(8k+5) - 1/(8k+6) ]
This formula still doesn't prove that Pi is normal, but it now links the distribution of digits in Pi to the field of chaotic dynamics. There is an unproved but plausible conjecture that certain sequences (just like the one described by the above formula) "uniformly dance in the limit between 0 and 1". This explains why the digits in Pi (and other constants, like log(2) or square root of 2) appear to be random, but doesn't prove that they actually are. However, if it is proved, the normality (in base 2) of Pi and many other mathematical constants will follow.