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Many people are interested in mathematics, or at least interested in the idea of being interested in math. But for too many people, they find math daunting. But it needn’t be so. Yes, there is jargon in math, like any other field. But many of the ideas in mathematics are simpler than you might realize, and also incredibly beautiful and elegant.
Well, Steven Strogatz has just released a wonderful gem of a book called The Joy of X, which is exactly about the wonder and beauty inherent in math. There is a joy to mathematics and Strogatz—a wonderfully gifted teacher and storyteller—takes us through the world of math, from the definition of a number all the way to calculus and probability theory.
Steve, also my graduate school adviser during my Ph.D., was kind enough do a Q&A via e-mail.
Samuel Arbesman: Your book is based on a celebrated series of columns at The New York Times which received an incredible response, far beyond the traditional math audience. What made you first want to tell the story of mathematics and relay it to a general audience in this series?
Steven Strogatz: It all got started when David Shipley, the op-ed editor for The New York Times, asked if I’d ever have time to write a series about math for his readers. He said he’d taken math in school, all the way up to calculus (and had even done pretty well in it) but he never really saw the point of it. He suspected many of his readers felt the same way. Could I give them some feeling for what math is all about and why it’s so captivating to those who get it? I thought it was a fantastic challenge.
Arbesman: Did you expect the response the series received?
Strogatz: No. But after the first piece appeared it became pretty clear that there was an audience for this sort of thing. That first article received over 500 comments and climbed to the top of the most e-mailed list. What was really surprising were the comments. One after another, people kept thanking me and the Times for what we were trying to do. That was the most frequent reaction: gratitude. It was tremendously encouraging.
Arbesman:
Why do you think so many people are frightened off by the world of math?
Strogatz: There seem to be a few different reasons. Many people I’ve talked to were humiliated or even traumatized by something in their math education. They liked math until the day they suddenly hit the wall. Maybe it was long division, or a terrifying algebra teacher, or proofs in geometry. After that they felt like they couldn’t get math anymore, and they began to think, "I’m not a math person."
I know that feeling myself. I had a linear algebra class in my freshman year of college that made me question whether I had the right stuff to be a math major. I felt a sense of dread before each test, and was completely bewildered by the professor. The textbook was even worse; it didn't have any pictures. Other kids in the class were raising their hands, shouting out the answers, but I was sitting there like a lump. It was very discouraging.
Other people tell me they aren’t afraid of math — they simply find it boring. And unfortunately it’s often taught that way, as a mechanical set of procedures. So of course a smart person would be turned off by it.
Arbesman: When did you first realize that mathematics was not simply another topic in school, but that there was a real pleasure in figuring how it explained the world?
Strogatz: It came in the form of an epiphany. In my first science class in high school, our teacher, Mr. diCurcio, asked us all to measure the period of a pendulum, the time it takes for the pendulum to swing back and forth. He gave each of us a little pendulum and a stopwatch. The pendulum could be lengthened or shortened in discrete clicks, and our assignment was to set the pendulum to a certain length, and then let it swing 10 times back and forth, meanwhile timing how long it took to make those 10 swings. Then we were supposed to repeat the measurement for pendulums of different lengths, and to plot our results as dots on a piece of graph paper, with the length of the pendulum on the x-axis and the time required for 10 swings on the y-axis. It was intended as a lesson in how to use graph paper, but it became clear to me after plotting the fourth or fifth dot that a pattern was emerging. The dots were falling on a curve. I recognized its shape because I’d seen it in algebra class. It was a parabola — the same shape that water makes when it comes out of a drinking fountain. I felt a peculiar chill, an enveloping sensation of fear and awe: this pendulum knows algebra! In that moment I suddenly understood what people meant by "laws of nature." It was a moment from which I've never really recovered. It felt like I was being let in on a secret. There were beautiful, hidden patterns in the world, patterns you couldn’t see unless you knew math.
Arbesman: One of my favorite examples you give in your book is a wonderfully elegant way of calculating the area of a circle using geometry. What is your favorite example of an elegant way to calculate something that might seem really complicated?
Strogatz: I love the proof that when you add consecutive odd numbers starting from 1, you always get a perfect square. Take 1+3. You get 4, which is 2 squared. Or add 1+3+5. Now you get 9, which is 3 squared. Same thing with 1+3+5+7; it’s 16, or 4 squared. There's a very clear pattern going on here that somehow connects sums of odd numbers to perfect squares. But why? It’s not so obvious. But if you look at it the right way, it becomes completely obvious. If you think of the odd numbers as L-shapes (for example, 7 can be drawn as a set of 7 circles, with 3 circles on the vertical side of the L and 3 circles on the bottom of the L, and a single circles in the corner), then when you add numbers like 1+3+5+7+9, you can see that you're just stacking these L-shapes together to make a square.
This argument is better than convincing; it's illuminating. That's what makes it "elegant."
Arbesman: There are so many great topics in the book, from prime numbers to group theory. How did you choose which topics to include in the book?
Strogatz: I tried to touch on all the greatest ideas of mathematics. To illustrate them I then picked topics that were appealing and important and that could be connected to something very accessible, like pop culture, sports, law, medicine, whatever.
For example, consider vector calculus. Unless you were a physics or math or engineering major in college, you probably never studied vector calculus. You might not even be sure what a vector is. And yet vectors are very fundamental in mathematics, a huge idea. To bring them down to earth, I introduce vectors by relating them to dance steps. Next comes the idea of addition of vectors, illustrated by how Pete Sampras hits a running forehand down the line. All of this ties into the larger story of why vector calculus is important – it allowed James Clerk Maxwell to figure out the laws of electricity and magnetism, and to set them in a mathematical framework that revealed that light is an electromagnetic wave. That's one of the great events in human history. It's right up there with Shakespeare and Einstein, Bach and da Vinci. Maxwell's electromagnetic waves, which he conjured up through vector calculus, ultimately led to television, radio, cell phones, and Wi-Fi. And it all starts with dance steps.
Arbesman: Each of your columns, as well as your book, was also read by a group of friends and colleagues and who provided feedback. These are all top-notch experts in math. How did they respond to your columns? Did anything surprise them?
Strogatz: I feel lucky to have such a brain trust, such a generous set of friends who happen to be geniuses as well as incredible nitpickers, which is exactly why I asked them to help. But as for whether anything surprised them, no, not much. They tended to know everything I was writing about already, of course. The only surprise I can think of came in a different venue, when I was recently interviewed on Science Friday. In response to a caller’s question, I sheepishly confessed that I had never learned how to check an arithmetic calculation by “casting out nines.” One of my friends thought this was pretty pathetic. He claimed to be “scandalized” by my admission and proceeded to school me on how it all works. That's the kind of guy you want on your team.
Arbesman: And do you have a favorite number (real, or imaginary) or a favorite equation?
Strogatz: I've always had a special fondness for 1/7. Its decimal expansion is .142857142857… I love that it repeats periodically but it takes so many digits to repeat itself. Better yet, it looks like it's trying to be part of the multiplication table for seven: 7 times 2 gives 14, the first two digits; then comes 28, which is 7 times 4; and finally comes 57, which looks likes it’s trying to be 7 times 8 but it isn’t quite. It's a fun pattern that I enjoyed as a kid.
But for deeper pleasures, how can you beat pi? On the one hand it symbolizes order — it's the embodiment of the most perfect shape we know, a circle. And on the other hand it's a symbol of mystery and disorder — no one has ever found a pattern in the digits of pi; they behave, in many respects, like a random sequence. So, to me, pi is the fairest of them all. It encapsulates the balance between order and chaos, and the infinite mystery of mathematics. What could be more beautiful?
