Even if you don’t usually read applied math books, you need to read Shape, by Jordan Ellenberg.
The subtitle tells you more than the title: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else. Yes, I know, it doesn’t sound much like your high-school geometry course. That’s because it isn’t.
Here’s the plan of the book. Or maybe it’s a diagram of Ellenberg’s mind:
If you prefer prose, read the beginning of the publisher’s summary on the book flap (normally I wouldn’t quote book flaps, but this is an exception):
How should a democracy choose its representatives? How can you stop a pandemic from sweeping the world? How to computers learn to play Go, and why is learning Go so much easier for them than learning to read a sentence? Can ancient Greek proportions predict the stock market? (Sorry, no.) What should your kids learn in school if they really want to learn to think? All these are questions about geometry. For real.
OK, so one aspect of this book is that it will expand your notion of geometry tenfold. Or more. If you’re not interested in one of the many topics that Ellenberg explains—some with a tight geometric connection, some with a much looser one—then just skip a chapter. You’ll lose a little continuity, but not too much.
Ellenberg needed an example of exponential growth for this book. Although it probably wasn’t in his original plans in the beforetimes, we have a good example now:
Every math teacher has longed for an example that will really bring home to students how exponential growth behaves. At the moment, unfortunately, we have one close at hand.
You know what he means. This is only one of several instances where Ellenberg makes it clear that he is not only a mathematician but also a math teacher (not the same, as I often remind my students). Here is another instance, this time on a more general issue:
I’ve been teaching math for more than twenty years now. When I started, I was driven by questions like this: What’s the right way to teach a mathematical concept? Examples first, then explanation? Explanation followed by examples? Letting students discover principles by examining the examples I present, or stating principles at the blackboard and letting students discover examples?
I’ve come to feel there’s no one right way. (Though there are certainly some wrong ways.) Different students are different and there is no One True Teaching Method that will ring everyone’s saliva bell.
I also like Ellenberg’s comments about “trial and error”—which my students insist on calling “guess and check,” perhaps because…well, let him explain:
Real math (like real life) is nothing like this. There’s a lot of trial and error. That method gets looked down on a lot, possibly because it has the word “error” in it. In math we’re not afraid of errors in it. In math we’re not afraid of errors. Errors are great! An error is just an opportunity to run another trial.
People in general like to be confirmed that they are right, which is another reason they are afraid of errors. In a different context I notice this tendency in Wheel of Fortune, where contestants will frequently waste a guess on the vowel “i” when trying to fill in a phrase containing a word that ends in “ng”: clearly they would get more information if they look elsewhere.
Near the end of the book we shift from specific applications of geometry to some general observations about it, observations that I try to emphasize when I’m teaching the subject:
There’s something special about geometry, something that makes it worth writing poems about. Everywhere else in the school curriculum, you must, in the end, defer to the teacher’s authority, or a textbook’s, when it comes to who fought in the French and Indian Wars, or what the principal products of Portugal are. In geometry you make your own knowledge. The power is in your hands.
That, of course, is exactly why geometry was correctly seen as dangerous by the Flatlanders and the Italian Jesuits. It represents an alternative source of authority. The Pythagorean Theorem isn’t true because Pythagoras said it was; it’s true because we can, ourselves, prove that it’s true. Behold!
And I have to inflict one more quote on you, the last paragraph of the book:
When we think, truly and deeply think, about geometric things—whether we are trying to chart the course of a pandemic, or walking through the tree of strategies that governs a game, or developing a working protocol for democratic representation, or understanding which things feel near to which other things, or trying to visualize the outside of the house from the inside, or, like Lincoln, rigorously criticizing our own beliefs and assumptions—we are, in a way, alone. But we’re alone together with everyone on earth. Everyone does geometry differently, but everyone does it. It is, just as its name says, the way we measure the world, and therefore (only in geometry do we get to say “therefore”) the way we measure ourselves.
Finally, let me point out that this book has great footnotes. The endnotes are also informative and useful, but the footnotes have flair and style. Also, by the way, as I pointed out in a recent blog post, Ellenberg appears (on screen!) in the movie Gifted, in a cameo performance as an MIT professor.
You still have to read the book to find out about democracy—from gerrymandering to proportional representation. It will be a pleasure.