Knuth’s “Why Pi” Talk at Stanford: Part 1

A few weeks ago I had another opportunity to meet a giant in my field: Donald Knuth was giving his annual Christmas Tree lecture at Stanford, where I happened to be near at the time. The most exciting part was probably right at the very beginning, when Knuth announced that, for the first (“and probably only”) time, he had finished two books on that same day: The Art of Computer Programming: Volume 4A, and Selected Papers on Fun and Games, with the pre-order links on Amazon going up the same day.

The topic of this year’s lecture was, according to the site:

Mathematicians have known for almost 300 years that the value of n! (the product of the first n positive integers) is approximately equal to, whereis the ratio of the circumference of a circle to its diameter. That’s “Stirling’s approximation.” But it hasn’t been clear why there is any connection whatever between circles and factorials; the appearance ofhas seemed to be purely coincidental, one of the amazing accidents of nature. Recently however Johan Wästlund has found a nice explanation of this phenomenon, and the reason turns out to be connected with the theory of trees.

I found the lecture originally hard to follow; although Knuth is now 74 years old and physically growing quite frail, his mind was still evidently much sharper than the audience’s; he was completely prepared for every question regarding extensions and corner cases of the material he was presenting. I considered the notes I took to be somewhat useless since I had spent so much time just trying to follow his reasoning. However, when the video became available, I had no excuse left, so I went through it again armed with a pen and the pause button, and I now have a set of notes somewhat worthy of posting here for anyone who missed it. Any errors or inconsistencies are almost definitely my fault and not his.

The notes themselves are after the jump.