Wednesday, March 14, 2012

Happy Pi Day!

Happy Pi Day. One of my heroes and one of the greatest minds of all time, Indian mathematician Srinivasa Ramanujan came up with this series for calculating Pi to a large number of decimal places in the early 20th century. To this day computers use variants of his series solutions to do the same thing but with much higher speed than Srinivasa could ever muster by hand.


Today there are many series that can converge on Pi rapidly and net you as many decimal places as you like. The current record is 2.7 Trillion digits. This is all just academic of course, there are no practical uses for accuracies of Pi greater than a half-dozen decimal points.


One novel solution is called the Bailey–Borwein–Plouffe (BBP) formula, which allows you to calculate a single specified digital bit of Pi without having to calculate all the previous bits. Just for fun, mathematicians have used the BBP to calculate the Quadrillionth bit of Pi (hint, it’s zero).


Underneath all this useless gameplay, Pi is a very useful number. It is used to calculate a great many geometrical problems…you can’t build a bridge or fly a rocket without Pi.


Along with other irrational numbers like the square root of 2, Pi gave no end of frustration to the Pythagoreans who thought that the universe was perfect and therefore should not have numbers that cannot be written out in full or represented by a ratio. Our language has trapped the word irrational in its modern meaning, but initially irrational meant simply that a number could not be expressed as a ratio. Ir-ratio-nal. Today, thanks to the Pythagoreans, it means something akin to illogical. To them perhaps it was, but we now know that the Pythagorean world-view was full of shit.


To give further mystique to our favorite constant, Pi was proven in the 1880’s to be transcendental as well. A transcendental number is a number, complex or otherwise, that is not the root of a non-constant polynomial equation with rational coefficients…in other words, Pi is a non-algebraic number. Too bad Pythagoras didn’t know that, it would have really pissed him off. I mean, thanks for the cool theorem, Pythagoras, but in your leisure time you pretty much hamstrung science for two thousand years. It wasn’t until Kepler came along in the 16th century and smashed through the tainted logic of the “Five perfect solids” that science got traction again. Bad Pythagoras. Bad boy.


There is a great Pi-related twist at the end of the Carl Sagan book Contact that hints that the basic structure of the universe may have been altered by very intelligent alien beings (or a god perhaps, but at that level the two would be indistinguishable to us) , who long-ago inserted a message in the form of a specific numerical pattern deep inside Pi, which is to say that they wrote the message into space-time itself and therefore into Pi, which is anchored to the geometry of the universe. This pattern was very deep (on the order of 10^20 decimal places) and only evident when Pi was calculated in Base-11, which itself is a jolting indicator that these beings would have been very different from us - we use Base-10 mathematics because we have 10 fingers…seriously, I kid you not…and computers use Base-2 because transistors have two states. No numerical base system is any better than any other, it’s a matter of preference. A being who used Base-11 math would probably have a very different physical form than we humans, if it had a physical form at all. They would have to be intelligent beyond both our imagination and our comprehension. In this case they had “timed” the release of this knowledge by placing the message deep enough into Pi that it required significant technology to find it. What it means, Carl never told us. This twist never made it into the movie, but it was brilliant and chilling.


And although Pi day doesn’t need any more street cred, Einstein was born on 3/14.


So happy 3.141592653589793238462643383279…

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