Sunday, March 4, 2018

Infinite lake surrounded by lighthouses: Why $1+1/4+1/9+\dots = \pi^2/6$

The Youtube channel 3Blue1Brown has made a wonderful video based on one of my papers! It's called The stunning geometry behind this surprising equation, and shows a geometrical proof of Leonhard Euler's amazing equation \[1 + \frac14 + \frac19 + \frac1{16} + \dots = \frac{\pi^2}6.\]
By the way, this means that a prediction from my very first blog post is coming true!

Several people notified me of this video already the same day it was posted (first of them was a student in my discrete math course), and it really is awesome!

 I have seen some videos from 3Blue1Brown before, but never checked out who makes them. This one was created by the team of Grant Sanderson and Ben Hambrecht. Grant Sanderson is the creator of 3Blue1Brown, and from their about-page I quote:

"...what excites me the most is finding that little nugget of explanation that really clarifies why something is true, not in the sense of a proof, but in the sense that you come away feeling that you could have discovered the fact yourself".

I wish I had said that myself. I mean I come away feeling that I could have!

About my personal relation to the Basel problem:

I remember thinking about the sum $1+1/4+1/9+\dots$ before I knew the answer. I asked one of my teachers about it, but he didn't know.

When I was something like 16 or 17 years old, I found a book in our school library called Högre matematik för poeter och andra matematiska oskulder (Higher mathematics for poets and other mathematical virgins). That book was my first encounter with Fourier series. I played around with the series for the saw-tooth and related functions, and used my pocket calculator to check the amazing results. Somehow I figured out that those series provided the answer to my question, and I was astonished to realize that it wasn't pi times something, it was pi squared!

I also realized I could integrate those series and sum the inverse powers whenever the exponent was even. I think I went as far as \[1+\frac1{2^{14}}+\frac1{3^{14}}+\dots = \frac{2\pi^{14}}{18243225},\] and used that to compute digits of pi on a computer in school.
I had a lot of fun discovering those things myself although I realized they had to be well-known. Later on I learned about the Riemann Zeta-function, Bernoulli numbers, and stuff.

By the way, and related to a discussion in the video, my friend Peter Grenholm from the years in Uppsala once said that "If there's a pi, there's a circle!". I think we were contemplating the mysteries of the normal distribution.

In 2004 I found an apparently new proof of the Wallis product formula, another amazing formula involving pi. That's another story, but while searching the literature to see if that proof was new (I still have doubts!) I stumbled upon a marvellous paper by the twin brothers Akiva and Isaak Yaglom from 1953. The paper was in Russian, but it was so short, elegant, and well written that I could understand it all by just looking at the equations. They derived three famous formulas by Wallis, Leibniz, and Euler, all involving pi. And they did it using trigonometric identities only, not calculus as in all the proofs I had seen before.

Those trigonometric identities were cute but still quite sophisticated, depending on complex numbers and a fair bit of algebra. But one day when I was teaching "Grunken" in Linköping, a sort of preparatory course for new students, one of the students asked me about a problem in their book. The problem was about expressing $\tan(x/2)$ in terms of $\tan x$. I knew there was such a formula, but hadn't really thought about it. But when we went through the solution, I immediately realized that the identities from the Yaglom-Yaglom paper could be easily derived, at least when a certain number $n$ was a power of 2. We discussed this in the coffee room in Linköping with some of my colleagues, and somebody, maybe Mats Aigner (?), showed me a paper by Josef Hofbauer where Euler's identity was derived in precisely this way (but Mats Aigner is not the M. Aigner cited in Hofbauer's paper, that's Martin Aigner).

So the idea was already known, but I still felt that there was something there that needed to be sorted out and simplified even further. If that trigonometry was so simple, why wouldn't the whole thing be just plain euclidean geometry?

So I tinkered with various ways of realizing the trigonometric identities geometrically. The problem was that in most of those constructions, the key quantities were tending either to zero or to infinity with increasing $n$. But finally I came up with a model that didn't need any annoying rescaling. I thought of it as stars revolving around a common center of gravity, and the relevant quantity was the total amount of light received at some other point in their orbit. But light-houses, as in the video, are arguably easier to move around!

I gave a talk about this at Chalmers in 2010, and put a LaTeXed version on my web page, mostly so that I wouldn't forget about it myself.

When I heard about the new video I didn't even remember which version of the paper I had put on my webpage. I had a different one where all the stars in the universe were on a line, and the theoretical physicists were debating the total amount of light. And then a little girl asked what would happen if the universe was finite and the stars were on a circle. A few years ago I had dinner with Lotta Olsson, an author of children's books, and we talked about poetry, mathematics, and Tönis Tönisson's book. Then I tried to rewrite the whole paper with rhymes in her style, but I didn't get it to work (yet!).

One of the funny things is that my paper isn't published anywhere except somewhere near the bottom of this webpage of mine that I only rarely update. It's not even on the arXiv. There is this industry (what should we call it?) in academia called bibliometry, where publications and citations are counted, but only in certain journals. A publication only counts as a "real" publication if a commercial publisher like Elsevier can trick universities to pay for it (using money from tax-payers of course). To enforce this corrupt system they talk about h-index, impact factors, and "outreach". This paper of mine has zero "outreach" of course, since it's not even a real paper according to this standard.

But somehow the internet tends to find what you put there. I have seen it mentioned on some blog somewhere, and I've been asked if I'm the author. And now, after a weekend, the 3Blue1Brown video explaining the paper has been viewed more than 200,000 times. I guess that means that most people who ever came across any of my mathematics did so in the last couple of days!

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