STEVEN STROGATZ
Steven Strogatz is a professor of applied mathematics at Cornell University. In 2007 he received the Communications Award, a lifetime achievement award for the communication of mathematics to the general public. He previously taught at the Massachusetts Institute of Technology, where he received the E.M. Baker Award, an institute-wide teaching prize selected solely by students. "Chaos," his series of 24 lectures on chaos theory, was filmed and produced in 2008 by The Teaching Company. He is the author, most recently, of “The Calculus of Friendship,” the story of his 30-year correspondence with his high school calculus teacher.
For Our Steven Strogatz Fans
By THE EDITORSProfessor Strogatz’s 15-part series on mathematics, which ran from late January through early May, is available on the “Steven Strogatz on the Elements of Math” page.
The Hilbert Hotel
By STEVEN STROGATZIn late February I received an e-mail message from a reader named Kim Forbes. Her six-year-old son Ben had asked her a math question she couldn’t answer, and she was hoping I could help:
Today is the 100th day of school. He was very excited and told me everything he knows about the number 100, including that 100 was an even number. He then told me that 101 was an odd number and 1 million was an even number, etc. He then paused and asked: “Is infinity even or odd?”
I explained that infinity is neither even nor odd. It’s not a number in the usual sense, and it doesn’t obey the rules of arithmetic. All sorts of contradictions would follow if it did. For instance, “if infinity were odd, 2 times infinity would be even. But both are infinity! So the whole idea of odd and even does not make sense for infinity.”
Kim replied:
Thank you. Ben was satisfied with that answer and kind of likes the idea that infinity is big enough to be both odd and even.
Although something got garbled in translation (infinity is neitherodd nor even, not both), Ben’s rendering hints at a larger truth. Infinity can be mind-boggling.
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Group Think
By STEVEN STROGATZMy wife and I have different sleeping styles — and our mattress shows it. She hoards the pillows, thrashes around all night long, and barely dents the mattress, while I lie on my back, mummy-like, molding a cavernous depression into my side of the bed.
Bed manufacturers recommend flipping your mattress periodically, probably with people like me in mind. But what’s the best system? How exactly are you supposed to flip it to get the most even wear out of it?
Brian Hayes explores this problem in the title essay of his recent book, “Group Theory in the Bedroom.” Double entendres aside, the “group” in question here is a collection of mathematical actions — all the possible ways you could flip, rotate or overturn the mattress so that it still fits neatly on the bed frame.
By looking into mattress math in some detail, I hope to give you a feeling for group theory more generally. It’s one of the most versatile parts of mathematics. It underlies everything from the choreography of contra dancing and the fundamental laws of particle physics, to the mosaics of the Alhambra and their chaotic counterparts like this image.
As these examples suggest, group theory bridges the arts and sciences. It addresses something the two cultures share — an abiding fascination with symmetry. Yet because it encompasses such a wide range of phenomena, group theory is necessarily abstract. It distills symmetry to its essence.
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Chances Are
By STEVEN STROGATZHave you ever had that anxiety dream where you suddenly realize you have to take the final exam in some course you’ve never attended? For professors, it works the other way around — you dream you’re giving a lecture for a class you know nothing about.
That’s what it’s like for me whenever I teach probability theory. It was never part of my own education, so having to lecture about it now is scary and fun, in an amusement park, thrill-house sort of way.
Perhaps the most pulse-quickening topic of all is “conditional probability” — the probability that some event A happens, given (or “conditional” upon) the occurrence of some other event B. It’s a slippery concept, easily conflated with the probability of B given A. They’re not the same, but you have to concentrate to see why. For example, consider the following word problem.
Before going on vacation for a week, you ask your spacey friend to water your ailing plant. Without water, the plant has a 90 percent chance of dying. Even with proper watering, it has a 20 percent chance of dying. And the probability that your friend will forget to water it is 30 percent. (a) What’s the chance that your plant will survive the week? (b) If it’s dead when you return, what’s the chance that your friend forgot to water it? (c) If your friend forgot to water it, what’s the chance it’ll be dead when you return?
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It Slices, It Dices
By STEVEN STROGATZMathematical signs and symbols are often cryptic, but the best of them offer visual clues to their own meaning. The symbols for zero, one and infinity aptly resemble an empty hole, a single mark and an endless loop: 0, 1, ∞. And the equals sign, =, is formed by two parallel lines because, in the words of its originator, Welsh mathematician Robert Recorde in 1557, “no two things can be more equal.”
In calculus the most recognizable icon is the integral sign:
Its graceful lines are evocative of a musical clef or a violin’s f-hole — a fitting coincidence, given that some of the most enchanting harmonies in mathematics are expressed by integrals. But the real reason that Leibniz chose this symbol is much less poetic. It’s simply a long-necked S, for “summation.”
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Change We Can Believe In
By STEVEN STROGATZLong before I knew what calculus was, I sensed there was something special about it. My dad had spoken about it in reverential tones. He hadn’t been able to go to college, being a child of the Depression, but somewhere along the line, maybe during his time in the South Pacific repairing B-24 bomber engines, he’d gotten a feel for what calculus could do. Imagine a mechanically controlled bank of anti-aircraft guns automatically firing at an incoming fighter plane. Calculus, he supposed, could be used to tell the guns where to aim.
Every year about a million American students take calculus. But far fewer really understand what the subject is about or could tell you why they were learning it. It’s not their fault. There are so many techniques to master and so many new ideas to absorb that the overall framework is easy to miss.
Calculus is the mathematics of change. It describes everything from the spread of epidemics to the zigs and zags of a well-thrown curveball. The subject is gargantuan — and so are its textbooks. Many exceed 1,000 pages and work nicely as doorstops.
But within that bulk you’ll find two ideas shining through. All the rest, as Rabbi Hillel said of the Golden Rule, is just commentary. Those two ideas are the “derivative” and the “integral.” Each dominates its own half of the subject, named in their honor as differential and integral calculus.
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Take It to the Limit
By STEVEN STROGATZIn middle school my friends and I enjoyed chewing on the classic conundrums. What happens when an irresistible force meets an immovable object? Easy — they both explode. Philosophy’s trivial when you’re 13.
But one puzzle bothered us: if you keep moving halfway to the wall, will you ever get there? Something about this one was deeply frustrating, the thought of getting closer and closer and yet never quite making it. (There’s probably a metaphor for teenage angst in there somewhere.) Another concern was the thinly veiled presence of infinity. To reach the wall you’d need to take an infinite number of steps, and by the end they’d become infinitesimally small. Whoa.
Questions like this have always caused headaches. Around 500 B.C., Zeno of Elea posed a set of paradoxes about infinity that puzzled generations of philosophers, and that may have been partly to blame for its banishment from mathematics for centuries to come. In Euclidean geometry, for example, the only constructions allowed were those that involved a finite number of steps. The infinite was considered too ineffable, too unfathomable, and too hard to make logically rigorous.
But Archimedes, the greatest mathematician of antiquity, realized the power of the infinite. He harnessed it to solve problems that were otherwise intractable, and in the process came close to inventing calculus — nearly 2,000 years before Newton and Leibniz.
In the coming weeks we’ll delve into the great ideas at the heart of calculus. But for now I’d like to begin with the first beautiful hints of them, visible in ancient calculations about circles and pi.
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Power Tools
By STEVEN STROGATZIf you were an avid television watcher in the 1980s, you may remember a clever show called “Moonlighting.” Known for its snappy dialogue and the romantic chemistry between its co-stars, it featured Cybill Shepherd and Bruce Willis as a couple of wisecracking private detectives named Maddie Hayes and David Addison. While investigating one particularly tough case, David asks a coroner’s assistant for his best guess about possible suspects. “Beats me,” says the assistant. “But you know what I don’t understand?” To which David replies, “Logarithms?” Then, reacting to Maddie’s look: “What? You understood those?”
(Click image to play clip.)
That pretty well sums up how many people feel about logarithms. Their peculiar name is just part of their image problem. Most folks never use them again after high school, at least not consciously, and are oblivious to the logarithms hiding behind the scenes of their daily lives.
The same is true of many of the other functions discussed in algebra II and pre-calculus. Power functions, exponential functions — what was the point of all that? My goal in this week’s column is to help you appreciate the function of all those functions, even if you never have occasion to press their buttons on your calculator.
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Think Globally
By STEVEN STROGATZThe most familiar ideas of geometry were inspired by an ancient vision — a vision of the world as flat. From parallel lines that never meet, to the Pythagorean theorem discussed in last week’s column, these are eternal truths about an imaginary place, the two-dimensional landscape of plane geometry.
Conceived in India, China, Egypt and Babylonia more than 2,500 years ago, and codified and refined by Euclid and the Greeks, this flat-earth geometry is the main one (and often the only one) being taught in high schools today. But things have changed in the past few millennia.
In an era of globalization, Google Earth and transcontinental air travel, all of us should try to learn a little about spherical geometry and its modern generalization, differential geometry. The basic ideas here are only about 200 years old. Pioneered by Carl Friedrich Gauss and Bernhard Riemann, differential geometry underpins such imposing intellectual edifices as Einstein’s general theory of relativity. At its heart, however, are beautiful concepts that can be grasped by anyone who’s ever ridden a bicycle, looked at a globe or stretched a rubber band. And understanding them will help you make sense of a few curiosities you may have noticed in your travels.
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Square Dancing
I bet I can guess your favorite math subject in high school.
It was geometry.
So many people I’ve met over the years have expressed affection for that subject. Arithmetic and algebra — not many takers there. But geometry, well, there’s something about it that brings a twinkle to the eye.
Is it because geometry draws on the right side of the brain, and that appeals to visual thinkers who might otherwise cringe at its cold logic? Maybe. But other people tell me they loved geometry precisely because it was so logical. The step-by-step reasoning, with each new theorem resting firmly on those already established — that’s the source of satisfaction for many. Read more…
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