In 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.
Let’s recall what we mean by pi. It’s a ratio of two distances. One of them is the diameter, the distance across the circle through its center. The other is the circumference, the distance around the circle. Pi is defined as their ratio, the circumference divided by the diameter.
If you’re a careful thinker, you might be worried about something already. How do we know that pi is the same number for all circles? Could it be different for big circles and little circles? The answer is no, but the proof isn’t trivial. Here’s an intuitive argument.
Imagine using a photocopier to reduce an image of a circle by, say, 50 percent. Then all distances in the picture — including the circumference and the diameter — would shrink in proportion by 50 percent. So when you divide the new circumference by the new diameter, that 50 percent change would cancel out, leaving the ratio between them unaltered. That ratio is pi.
Of course, this doesn’t tell us how big pi is. Simple experiments with strings and dishes are good enough to yield a value near 3, or if you’re more meticulous, 3 and 1/7th. But suppose we want to find pi exactly or at least approximate it to any desired accuracy. What then? This was the problem that confounded the ancients.
Before turning to Archimedes’s brilliant solution, we should mention one other place where pi appears in connection with circles. The area of a circle (the amount of space inside it) is given by the formula
Here A is the area, π is the Greek letter pi, and ris the radius of the circle, defined as half the diameter.
All of us memorized this formula in high school, but where does it come from? It’s not usually proven in geometry class. If you went on to take calculus, you probably saw a proof of it there, but is it really necessary to use calculus to obtain something so basic?
Yes, it is.
What makes the problem difficult is that circles are round. If they were made of straight lines, there’d be no issue. Finding the areas of triangles, squares and pentagons is easy. But curved shapes like circles are hard.
The key to thinking mathematically about curved shapes is to pretend they’re made up of lots of little straight pieces. That’s not really true, but it works … as long as you take it to the limit and imagine infinitely many pieces, each infinitesimally small. That’s the crucial idea behind all of calculus.
Here’s one way to use it to find the area of a circle. Begin by chopping the area into four equal quarters, and rearrange them like so.
The strange scalloped shape on the bottom has the same area as the circle, though that might seem pretty uninformative since we don’t know its area either. But at least we know two important facts about it. First, the two arcs along its bottom have a combined length of πr, exactly half the circumference of the original circle (because the other half of the circumference is accounted for by the two arcs on top). Second, the straight sides of the slices have a length of r, since each of them was originally a radius of the circle.
Next, repeat the process, but this time with eight slices, stacked alternately as before.
The scalloped shape looks a bit less bizarre now. The arcs on the top and the bottom are still there, but they’re not as pronounced. Another improvement is the left and right sides of the scalloped shape don’t tilt as much as they used to. Despite these changes, the two facts above continue to hold: the arcs on the bottom still have a net length of πr, and each side still has a length of r. And of course the scalloped shape still has the same area as before — the area of the circle we’re seeking — since it’s just a rearrangement of the circle’s eight slices.
As we take more and more slices, something marvelous happens: the scalloped shape approaches a rectangle. The arcs become flatter and the sides become almost vertical.
In the limit of infinitely many slices, the shape is a rectangle. Just as before, the two facts still hold, which means this rectangle has a bottom of width πr and a side of height r.
But now the problem is easy. The area of a rectangle equals its width times its height, so multiplying πr times r yields an area of πr2 for the rectangle. And since the rearranged shape always has the same area as the circle, that’s the answer for the circle too!
What’s so charming about this calculation is the way infinity comes to the rescue. At every finite stage, the scalloped shape looks weird and unpromising. But when you take it to the limit — when you finally “get to the wall” — it becomes simple and beautiful, and everything becomes clear. That’s how calculus works at its best.
Archimedes used a similar strategy to approximate pi. He replaced a circle by a polygon with many straight sides, and then kept doubling the number of sides to get closer to perfect roundness. But rather than settling for an approximation of uncertain accuracy, he methodically bounded pi by sandwiching the circle between “inscribed” and “circumscribed” polygons, as shown below for 6-, 12- and 24-sided figures.
Then he used the Pythagorean theorem to work out the perimeters of these inner and outer polygons, starting with the hexagon and bootstrapping his way up to 12, 24, 48 and ultimately 96 sides. The results for the 96-gons enabled him to prove that
In decimal notation (which Archimedes didn’t have), this means pi is between 3.1408 and 3.1429.
This approach is known as the “method of exhaustion” because of the way it traps the unknown number pi between two known numbers that squeeze it from either side. The bounds tighten with each doubling, thus exhausting the wiggle room for pi.
In the limit of infinitely many sides, both the upper and lower bounds would converge to pi. Unfortunately, this limit isn’t as simple as the earlier one, where the scalloped shape morphed into a rectangle. So pi remains as elusive as ever. We can discover more and more of its digits — the current record is over 2.7 trillion decimal places — but we’ll never know it completely.
Aside from laying the groundwork for calculus, Archimedes taught us the power of approximation and iteration. He bootstrapped a good estimate into a better one, using more and more straight pieces to approximate a curved object with increasing accuracy.
More than two millennia later, this strategy matured into the modern field of “numerical analysis.” When engineers use computers to design cars to be optimally streamlined, or when biophysicists simulate how a new chemotherapy drug latches onto a cancer cell, they are using numerical analysis. The mathematicians and computer scientists who pioneered this field have created highly efficient, repetitive algorithms, running billions of times per second, that enable computers to solve problems in every aspect of modern life, from biotech to Wall Street to the Internet. In each case, the strategy is to find a series of approximations that converge to the correct answer as a limit.
And there’s no limit to where that’ll take us.
- The history and intellectual legacy of Zeno’s paradoxes are discussed in: J. Mazur, Zeno’s Paradox (Plume, 2008).
- For a delightfully opinionated and witty history of pi, see: P. Beckmann, A History of Pi (St. Martin’s Press, 1976).
- Bill Willis at Worsley School OnLine has given a very clear explanation of how to find the area of the circle, using the same argument as above but fleshed out in more detail. The school website contains many other excellent math and science resourcesfor students, teachers and parents.
- The PBS television series “Nova” ran a wonderful episode about Archimedes, infinity and limits called “Infinite Secrets.” It originally aired on Sept. 30, 2003. The program website includes many online resources, including the program transcript and interactive demonstrations.
- For readers wishing to see the mathematical details of Archimedes’s method of exhaustion, Neal Carothers has used trigonometry (equivalent to the Pythagorean gymnastics that Archimedes relied on) to derive the perimeters of the inscribed and circumscribed polygons between which the circle is trapped. Peter Alfeld’s website features an interactive Java applet that lets you change the number of sides in the polygons.
- The individual steps in Archimedes’s original argument are of historical interest but you might find them disappointingly obscure.
- Anyone curious about the heroic computations of pi to enormous numbers of digits should enjoy Richard Preston’s profile of the Chudnovsky brothers. Entitled “The Mountains of Pi,” this affectionate and surprisingly comical piece appeared in the Mar. 2, 1992, issue of The New Yorker, and more recently as a chapter in: R. Preston, Panic in Level Four (Random House, 2008).
- For a textbook introduction to the basics of numerical analysis, see: W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes, 3rd ed. (Cambridge University Press, 2007).
Thanks to Tim Novikoff and Carole Schiffman for their comments and suggestions, and to Margaret Nelson for preparing the illustrations.
Editor’s Note: A correction was made to an earlier version of this column, to fix a misspelling of the name of the publisher of Zeno’s Paradox.
Source: Take It to the Limit – The New York Times