2013-09-30
The depiction of mathematics in popular science often falls woefully short of reality. Authors tend to prance around the "beauty" of numbers like e, π, and the golden ratio φ, but in all honesty numbers are only one part of mathematics, and they're not even the best part.
In a series of short vignettes, I want to take a quick dip into some mind-bending ideas that mathematicians enjoy (and use as tools) on a daily basis. Hopefully you can experience the same kind of joy I did when I first learned about these topics, and with luck you just might be enticed enough to dive deeper.
While there are many different styles of doing mathematics (I plan to write a separate article about this soon), one thing that all mathematicians agree on is the value of a good proof. It's weird to think of proofs as having an aesthetic quality, but they do. Good proofs glean insight into why something is true, and they often involve new techniques that can be used to prove other things. And most of all, good proofs are beautiful. When mathematicians talk about "the beauty of π", they don't mean that the number itself is beautiful. It's just a number and is no more or less beautiful than 2π or -π or 1/π. What is beautiful is how π relates to so many other things in mathematics, and these relationships are established by interesting proofs.
And much like the career of an artist is in the compositions he or she creates, proofs are the brushstrokes of mathematics. No idea is interesting unless there are proofs of theorems that show it is interesting. We are simply not happy with a true fact, as a physicist might be if he accidentally stumbled upon some physical law that was empirically accurate. No, a mathematician would need to prove that the law is consistent and follows logically from natural assumptions. Without a proof, "laws" are just words in the wind.
I want to show you the aesthetics of a proof with two simple examples, and I assure you that you can understand them without any mathematical knowledge. While it might not be immediately obvious how to solve the problems, the proofs are so simple and obvious a seven-year-old could understand them. Indeed, this is a big part of the beauty of these particular problems.
The first problem is nice because, as my colleague Maxwell mentioned, you can explain it over a dinner table without any paper or a pencil. It's simply a shift in perspective that makes an initially opaque question crystal clear.
Suppose you have a tournament of 16 people playing table tennis. The tournament is single-elimination, so the winner of a game goes on while the loser does not. The question is: how many games are played in total?
In the first round, there are eight games, and eight winners. In the second round, there are four games and four winners. In the third round, two games and two winners, and in the last round there is one game and one winner. With some elbow grease or a calculator, you could figure out that 15 games are played in total. Sweet.
But in this tournament we got lucky because everyone played a game in every round. Nobody had to sit out because we got lucky with the initial number of players. Now what if there were 1,000 players? How many games would be played then?
The same analysis of counting how many games there are in each round becomes much more difficult, because you have to keep track of how many players are sitting out in each round. If I tried for a moment to write out the calculations on paper, I'd quickly fill up the page with illegible chicken scratch. I might even get an answer at the end: 999 games. That's a bit weird: it's just the number of players minus one, which was true when there were 16 players, too. I'd have to check and double check that I didn't make any mistakes, but even if my counting was sound I don't really understand why I got this answer, nor am I sure that it will work if I change the starting number of players. All in all, I have little more insight into the problem than when I started. Might there be a better way?
Indeed, here comes the beautiful proof. We just notice that every player loses exactly one game, except for the winner of the whole tournament. So if we want to count the number of games, we can just count the number of losers, which is obviously 999. And this idea will not change if the number of players changes. Sha-zam!
If you've read the previous post in this series, you might recognize this as a bijection between the set of games played and the set of losers. Indeed, coming up with clever bijections is a standard mathematical tool used to count things. And so we have a beautiful solution to this initially difficult-sounding problem, and a new method of proof we can try to use on any other counting problem we might be faced with.
I've told this proof to mathematicians and English majors and kids alike, and it seems to have a widespread aesthetic appeal. And at the same time, it's not even immediately clear to non-mathematicians that this is mathematics at all! It doesn't involve derivatives or polynomials or fractions or the other things most people assume mathematics is limited to.
Here's an even simpler example, which is often cited as the simplest example of a beautiful proof. Say you have a triangle sitting inside a rectangle like this:
How much space do you think this triangle takes up within the rectangle? Maybe 1/3? Maybe 1/2? Maybe a little less than 2/3? You can guess and measure all you want, but how can you be entirely sure that you're right?
Well here comes the inspired proof: we can simply draw a line:
Explaining this is trivial. The triangle takes up exactly half of the rectangle because it does so on both sides of the line. We know it takes up half on both sides of the line because of the symmetry of each piece. It is completely obvious, but we can say it in words: rotating something (the piece of the triangle) preserves its area, and the part of the rectangle to the left of the line is just the piece of the triangle and the piece after rotating it 180 degrees.
Here the deep tool we're using is symmetry: drawing new lines help us decompose complicated shapes into easy-to-manage pieces, and using nice rotations and reflections we can reason about the pieces.
The beauty in both the tournament proof and the triangle proof lies in simplicity, cleverness, and the discovery of a deep new tool. If we wanted to ask new questions about different kinds of tournaments (say double-elimination) or different kinds of triangles (say, if the peak is shifted left outside the rectangle), we can use these new techniques to attack them. If we're lucky, we can come up with a more general proof that applies to all tournament problems (or triangle problems) that we could encounter. This is called a classification theorem, and finding such a proof is a monumental accomplishment.
Want to see some more beautiful proofs? I made a gallery of some of my favorites over the years on my blog Math ∩ Programming. One that is particularly good for the non-mathematical crowd is the one about tiling a chessboard with dominoes. A more advanced elegant geometry proof involves spontaneously turning circles into cones. There is also a great picture proof about the number of ways to choose 2 things from a collection of n things. Enjoy!