Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

Feynman asked Wouk if he knew calculus. No, Wouk admitted, he didn’t. “You had better learn it,” said Feynman. “It’s the language God talks.”

But why should the universe respect the workings of any kind of logic, let alone the kind of logic that we puny humans can muster? This is what Einstein marveled at when he wrote, “The eternal mystery of the world is its comprehensibility.” And it’s what Eugene Wigner meant in his essay “On the Unreasonable Effectiveness of Mathematics in the Natural Sciences” when he wrote, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”

When we had only four slices, or eight, or sixteen, the best we could do was rearrange the pizza into an imperfect scalloped shape. After an unpromising start, the more slices we took, the more rectangular the shape became. But it was only in the limit of infinitely many slices that it became truly rectangular. That’s the big idea behind calculus. Everything becomes simpler at infinity.

Planck length = √ ħG / c3 . When we plug in the measured values of G, ħ, and c, the Planck length comes out to be about 10–35 meters, a stupendously small distance that’s about a hundred million trillion times smaller than the diameter of a proton. The corresponding Planck time is the time it would take light to traverse this distance, which is about 10–43 seconds. Space and time would no longer make sense below these scales. They’re the end of the line. These numbers put a bound on how fine we could ever slice space or time.

Other stories cast him as a military magician, a warrior-scientist / one-man death squad. According to these legends, when his home city of Syracuse was under siege by the Romans in 212 BCE, Archimedes—by then an old man, around seventy—helped defend the city by using his knowledge of pulleys and levers to make fantastical weapons, “war engines” such as grappling hooks and giant cranes that could lift the Roman ships out of the sea and shake the sailors from them like sand being shaken out of a shoe. As Plutarch described the terrifying scene, “A ship was frequently lifted up to a great height in the air (a dreadful thing to behold), and was rolled to and fro, and kept swinging, until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall.”

We will never know all the digits of pi. Nevertheless, those digits are out there, waiting to be discovered. As of this writing, twenty-two trillion digits have been computed by the world’s fastest computers. Yet twenty-two trillion is nothing compared to the infinitude of digits that define the actual pi. Think of how philosophically disturbing this is. I said that the digits of pi are out there, but where are they exactly? They don’t exist in the material world. They exist in some Platonic realm, along with abstract concepts like truth and justice.

With its yin and yang binaries, pi is like all of calculus in miniature. Pi is a portal between the round and the straight, a single number yet infinitely complex, a balance of order and chaos.

In October 1998 a battered medieval prayer book came up for auction at Christie’s and sold to an anonymous private collector for $2.2 million. Barely visible under its Latin prayers lay faint geometrical diagrams and mathematical text written in tenth-century Greek. The book is a palimpsest; in the thirteenth century, its parchment folios had been washed and scraped clean of the original Greek and overwritten with Latin liturgical text. Fortunately, the Greek was not completely obliterated. It contains the only surviving copy of Archimedes’s Method.

The characters in films like Shrek, Finding Nemo, and Toy Story seem lifelike and real, in part because they embody an Archimedean insight: Any smooth surface can be convincingly approximated by triangles.

Just as Archimedes used a mosaic of infinitely many triangular shards to represent a segment of a smoothly curved parabola, modern-day animators at DreamWorks created Shrek’s round belly and his cute little trumpet-like ears out of tens of thousands of polygons. Even more were required for a tournament scene in which Shrek battled local thugs; each frame of that scene took over forty-five million polygons.

For GPS, it works like this: When the signals from the four satellites arrive at the receiver, your GPS gadget compares the time they were received to the time they were transmitted. Those four times are all slightly different, because the satellites are at four different distances away from you. Your GPS device multiplies those four tiny time differences by the speed of light to calculate how far away you are from the four satellites overhead. Because the positions of the satellites are known and controlled extremely accurately, your GPS receiver can then triangulate those four distances to determine where it is on the surface of the Earth. It can also figure out its elevation and speed. In essence, GPS converts very precise measurements of time into very precise measurements of distance and thereby into very precise measurements of location and motion.

Moreover, he found that for every planet, the sun was located at one of the foci of the planet’s elliptical orbit.

If we look at Mars at any two points in its orbit separated by equal amounts of time, the resulting sectors will always have equal areas, no matter where they are in the orbit. In a nutshell, the second law says that the planets do not move at a constant speed. Instead, the closer they get to the sun, the faster they move. The statement about equal areas in equal times is a way of making this precise.

In the preface to Harmonies of the World (1619), he wrote in ecstasy about finally seeing the pattern in God’s plan: “Now, since the dawn eight months ago, since the broad daylight three months ago, and since a few days ago, when the full sun illuminated my wonderful speculations, nothing holds me back. I yield freely to the sacred frenzy.” The numerological pattern that enraptured Kepler was his discovery that the square of the period of revolution of a planet is proportional to the cube of its average distance from the sun. Equivalently, the number T2/a3 is the same for all the planets. Here, T measures how long it takes a planet to go around the sun once (a year for the Earth, 1.9 years for Mars, 11.9 years for Jupiter, and so on), while a measures how far away the planet is from the sun. That’s a bit tricky to define, because the actual distance changes from week to week as a planet moves in its elliptical orbit; sometimes it’s closer to the sun and sometimes it’s farther away. To account for this effect, Kepler defined a as the average of the planet’s nearest and farthest distances to the sun. The

Kepler’s obsession with geometry, both sacred and profane, verged on the irrational. But his fervor made him who he was. As the writer Arthur Koestler astutely observed, “Johannes Kepler became enamored with the Pythagorean dream, and on this foundation of fantasy, by methods of reasoning equally unsound, built the solid edifice of modern astronomy. It is one of the most astonishing episodes in the history of thought, and an antidote to the pious belief that the Progress of Science is governed by logic.”

From a modern perspective, there are two sides to calculus. Differential calculus cuts complicated problems into infinitely many simpler pieces. Integral calculus puts the pieces back together again to solve the original problem.

Given that cutting comes naturally before rebuilding, it seems sensible for a novice to learn differential calculus first. And indeed, that’s how all calculus courses begin today. They start with derivatives—the relatively easy techniques for slicing and dicing—and then work their way up to integrals, the much harder techniques for reassembling the pieces into an integrated whole. Students find it more comfortable to learn calculus in this order because the easier material comes first. Their teachers like it because the subject seems more logical this way. Yet, strangely enough, history unfolded in the opposite order. Integrals were already in full swing in ancient Greece in Archimedes’s work around 250 BCE, whereas derivatives weren’t even a gleam in anybody’s eye until the 1600s. Why did differential calculus—the easier side of the subject—develop so much later than integral calculus? It’s because differential calculus grew out of algebra, and algebra took centuries to mature, migrate, and mutate. In its original form in China, India, and the Islamic world, algebra was entirely verbal. Unknowns were words, not today’s x and y. Equations were sentences, and problems were paragraphs. But soon after algebra arrived in Europe, around 1200, it evolved into an art of symbols. That made algebra more abstract . . . and more powerful. This new breed, symbolic algebra, then coupled with geometry and spawned an even stronger hybrid, analytic geometry, which in turn begat a zoo of n...

In 1650, he reluctantly took a position in Sweden (which he scorned as “the country of bears, amid rocks and ice”) as Queen Christina’s personal philosophy tutor. Unfortunately for Descartes, the energetic young queen was an early riser. She insisted on lessons at five in the morning, an ungodly hour for anyone but especially for Descartes, accustomed to getting up at noon his whole life. That winter in Stockholm was the coldest in decades. After a few weeks, Descartes caught pneumonia and died.

In analysis, one solves a problem by starting at the end, as if the answer had already been obtained, and then works back wishfully toward the beginning, hoping to find a path to the given assumptions. It’s what kids in school think of as working backward from the answer to figure out how to get there. Synthesis goes in the other direction. It starts with the givens, and then, by stabbing in the dark, trying things, you are somehow supposed to move forward to a solution, step by logical step, and eventually arrive at the desired result. Synthesis tends to be much harder than analysis because you don’t ever know how you’re going to get to the solution until you do. The ancient Greeks regarded synthesis as carrying more logical force, more persuasive power, than analysis. Synthesis was considered the only valid way to prove a result; analysis was a practical way to find the result. If you wanted a rigorous demonstration, you had to do synthesis. That’s why, for example, Archimedes used his analytical method of balancing shapes on seesaws to find his theorems but then switched to the synthetic method of exhaustion to prove them.

A million seconds is a little under two weeks; a billion seconds is about thirty-two years.

Given that 90 is a little less than 100, and 100 equals 102, it seems like 90 should equal 10 raised to some number slightly less than 2. But raised to what number, exactly? Logarithms were invented to answer such questions. On a calculator, if you type in 90 and then press the log button, you get log 90 = 1.9542 . . . . That’s the answer: 101.9542 . . . = 90. In this way, logarithms enable us to write any positive number as a power of ten. Doing that makes many calculations easier and also reveals surprising connections between numbers. Look what happens if we multiply 90 by a factor of 10 or 100 and then take its log again: log 900 ≈ 2.9542 . . . and log 9000 = 3.9542 . . . .

logarithms also describe many natural phenomena. For example, our perception of pitch is approximately logarithmic. When a musical pitch goes up by successive octaves, from one do to the next, that increase corresponds to successive doublings of the frequency of the associated sound waves. Yet although the waves oscillate twice as fast for every octave increase, we hear the doublings—which are multiplicative changes in frequency—as equal upward steps in pitch, meaning equal additive steps. It’s freaky. Our minds fool us into believing that 1 is as far from 2 as 2 is from 4, and as 4 is from 8, and so on. We somehow sense frequency logarithmically.

This natural base is called e. It’s a number close to 2.718 (I’ll explain where it comes from in a minute), but its numerical value is beside the point. The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of ex is ex itself. This marvelous property simplifies all calculations about exponential functions when they are expressed in base e. No other base enjoys this simplicity.

remarkable property of sine waves, namely, when a variable follows a perfect sine-wave pattern, its rate of change is also a perfect sine wave timed a quarter of a cycle ahead. This self-regeneration property is unique to sine waves. No other kinds of waves have it. It could even be taken as a definition of sine waves. In that sense, our data hint at a marvelous phenomenon of rebirth inherent in perfect sine waves.

Another example comes from a simplified model of predator-prey interactions in biology. Imagine a population of sharks preying on a population of fish. When the fish are at their maximum population level, the shark population grows at its maximum rate because there are so many fish to eat. The shark population continues to climb and reaches its own maximum level a quarter cycle later, by which time the fish population has dropped, having been preyed on so severely a quarter cycle earlier. An analysis of this model shows that the two populations oscillate out of phase by 90 degrees. Similar predator-prey oscillations are seen elsewhere in nature, for example, in annual fluctuations of Canadian hare and lynx populations as recorded by fur-trapping companies in the 1800s (though the real explanation for those oscillations is undoubtedly more complicated, as is often the case in biology).

How fast did he run? Well, 100 meters in 9.69 seconds translates to 100/9.69 = 10.32 meters per second. In more familiar units, that’s about 37 kilometers per hour, or 23 miles per hour. But that was his average speed over the whole race. He went slower than that at the beginning and end and faster than that in the middle. More detailed information is available from his split times recorded every 10 meters down the track. He covered the first 10 meters in 1.83 seconds, corresponding to an average speed of 5.46 meters per second there. His fastest splits occurred at 50 to 60 meters, 60 to 70 meters, and 70 to 80 meters. He blazed through those 10-meter sections in 0.82 seconds each, for an average speed of 12.2 meters per second. In the final 10 meters, when he eased up and broke form, he decelerated to an average speed of 11.1 meters per second.

If we try to push the resolution of our measurements too far, if we look at any phenomenon in excruciatingly fine detail in time or space, we will start to see a breakdown of smoothness. In Usain Bolt’s speed data, the wiggles took a smooth trend and made it look as bushy as a pipe cleaner. The same thing would happen with any form of motion if we could measure it at the molecular scale. Down at that level, motion becomes jittery and far from smooth. Calculus would no longer have much to tell us, at least not directly.

Quantum foam

“Art,” said Picasso, “is a lie that makes us realize truth.”

the word calculus itself has stories to tell. It comes from the Latin root calx, meaning a small stone, a reminder of a time long ago when people used pebbles for counting and thus for calculations. The same root gives us words like calcium, chalk, and caulk. Your dentist might use the word calculus to refer to that gunk on your teeth, the tiny pebbles of solidified plaque the hygienist scrapes off when you go for a cleaning. Doctors use the same word for gallstones, kidney stones, and bladder stones. In a cruel irony, both Newton and Leibniz, the pioneers of calculus, died in excruciating pain while suffering from calculi—a bladder stone for Newton, a kidney stone for Leibniz.

Just as acceleration is the rate of change of speed, speed is the rate of change of distance.

tangent lines per se were not that important. More important than such lines were their slopes, for it was slopes that led to the concept of the derivative. As we saw in the last chapter, the derivative arose very naturally in geometry as the slope of a curve. And derivatives also arose in physics as other rates of change, such as speeds. Thus, derivatives suggested a link between slopes and speeds and, more broadly, between geometry and motion.

ponder the diagram above and imagine sliding x to the right at a constant speed. You could even think of x as time; Newton often did. Then the area of the gray region changes continuously as x moves. Because that area depends on x, it should be regarded as a function of x, so we write it as A(x). When we want to stress that this area is a function of x (as opposed to a frozen number), we refer to it as the area accumulation function, or sometimes just the area function.

what’s the rate at which paint is being laid down when the roller is at x? To answer that, think about what happens in the next infinitesimal interval of time. The roller rolls to the right through some infinitesimal distance dx. Meanwhile, as it traverses that tiny distance, it keeps its length y in the vertical direction almost perfectly constant, since there’s almost no time for it to change its length during the infinitesimally brief roll (a fine point that we’ll discuss in the next chapter). During that brief interval, it paints what is essentially a tall, thin rectangle of height y, infinitesimal width dx, and infinitesimal area dA = y dx. Dividing this equation by dx then reveals the rate at which area accumulates.

dA⁄dx = y. This tidy formula says that the total painted area under the curve increases at a rate given by the current length y of the paint roller. It makes sense; the longer the roller currently is, the more paint it lays down in the next instant, and so the faster the area accumulates.

A(x) derivative→ y(x) derivative→ dy/dx area under curve curve slope of curve

The derivative of the area gives the function/curve, and the derivative of thr function gives the slope of that function

the area A and the curve y are also related by a derivative—the fundamental theorem has revealed that the derivative of A is y.

the fundamental theorem doesn’t fully solve the area problem. It provides information about the rate at which the area changes, but we still need to infer the area itself.

An area is, from an Archimedean standpoint, an infinite sum of infinitesimal rectangular strips. As such, an area is an integral. It’s the integrated collection of all the pieces put back together, an accumulation of infinitesimal change.

the area problem is not just about area. It’s not just about shape or the relationship between distance and speed or anything that narrow. It’s completely general. From a modern perspective, the area problem is about predicting the relationship between anything that changes at a changing rate and how much that thing builds up over time. It’s about the fluctuating inflow to a bank account and the accumulated balance of money in it. It’s about the growth rate of the world’s population and the net number of people on Earth. It’s about the changing concentration of a chemotherapy drug in a patient’s blood and the accumulated exposure to that drug over time. That total exposure affects how potent the chemo will be, as well as how toxic. Area matters because the future matters.

When little Isaac was three, his mother, Hannah, remarried and left him in the care of his maternal grandparents. (His mother’s new husband, Reverend Barnabas Smith, insisted on this arrangement; he was a wealthy man twice her age and wanted a young wife but not a young son.) Understandably, Isaac resented his stepfather and felt abandoned by his mother. Later in life, on a list of sins he’d committed before the age of nineteen, he included this entry: “13. Threatning my father and mother Smith to burne them and the house over them.” The next entry was darker: “14. Wishing death and hoping it to some.” And then this: “15. Striking many. 16. Having uncleane thoughts words and actions and dreamese.”

Consider the simplest, bleakest situation imaginable: an isolated body alone in an empty universe. How would it move? Well, since there’s nothing around to push it or pull it, the force on the body is zero: F = 0. Then, since m is not zero (assuming the body has some mass), Newton’s law yields F/m = a = 0, which implies that dv/dt = 0 as well. But dv/dt =0 means the lonesome body’s velocity doesn’t change during the infinitesimal time interval dt. Nor does it change during the next interval, or the one after that. The upshot is that when F = 0, a body maintains its velocity forever. This is Galileo’s principle of inertia: In the absence of an outside force, a body at rest stays at rest, and a body in motion stays in motion and moves at a constant velocity. Its speed and direction never change. We have just deduced the law of inertia as a logical consequence of Newton’s deeper law of motion, F = ma.

Force, he realized, was not needed to produce motion; it was needed to produce changes in motion. It was force that was responsible for making bodies speed up, slow down, or depart from a straight path.

His idea was that the moon’s orbit was a kind of never-ending fall to the Earth. But unlike a falling apple, the falling moon doesn’t crash to the ground because it’s also simultaneously cruising sideways due to inertia. It’s like one of Galileo’s cannonballs, gliding sideways and falling at the same time, tracing a curved path, except that it’s gliding so fast that it never reaches the surface of the spherical Earth curving away beneath it. As its orbit deviates from a straight line, the moon accelerates—not in the sense that its speed changes, but its direction of motion changes. What pulls it off a straight-line path is the incessant tug of the Earth’s gravity. The resulting type of acceleration is called centripetal acceleration, a tendency to be pulled toward a center—in this case, the center of the Earth.

The amplitudes of the optimal sine waves follow a prescription that Fourier discovered: Triangle wave = sinx − 1⁄9sin3x + 1⁄25sin5x − 1⁄49sin7x + · · · . This infinite sum is called the Fourier series for the triangle wave. Notice the cool numerical patterns in it. Only odd frequencies 1, 3, 5, 7, . . . appear in the sine waves, and their corresponding amplitudes are the inverse squares of the odd numbers with alternating plus and minus signs. Unfortunately, I can’t easily explain why this prescription works; we would have to plow through too much nitty-gritty calculus to see where those magic amplitudes come from. But the point is that Fourier knew how to compute them. By doing so he was able to synthesize a triangle wave or any other arbitrarily complicated curve out of much simpler sine waves. Fourier’s big idea is the basis for music synthesizers.

overtones. That’s the musical term for the waves like sin 3x and sin 5x in the earlier formula for the triangle wave. Overtones add color to a note by adding in multiples of the fundamental frequency. In addition to the sine wave at 440 cycles a second, a synthesized triangle wave includes a sine-wave overtone at three times that frequency (3 × 440 = 1320 cycles per second). That overtone is not as strong as the fundamental sin x mode. Its relative amplitude is only 1/9 as large as the fundamental, and the other odd-numbered modes are even weaker. In musical terms, these amplitudes determine the loudness of the overtones. The richness of the sound of a violin has to do with its particular combination of softer and louder overtones.

The unifying power of Fourier’s idea is that the sound of any musical instrument can be synthesized by an array of infinitely many tuning forks. All we need to do is strike the tuning forks with the right strengths and at the right times and, incredibly, out pops the sound of a violin or a piano or even a trumpet or an oboe, although we’re using nothing more than colorless sine waves. This is essentially how the first electronic synthesizers worked: they reproduced the sound of any instrument by combining a large number of sine waves.

To visualize the vibrations, Chladni sprinkled a fine dust of sand onto the plate before he bowed it. When he stroked the plate, the sand bounced off the parts that were vibrating the most and settled in the parts that weren’t vibrating at all. The resulting curves are now called Chladni patterns.

For a standard microwave oven that runs at 2.45 GHz (meaning the waves vibrate back and forth 2.45 billion times a second),

you can use your oven to calculate the speed of light. Multiply the frequency of vibration (listed on the oven’s door frame) by the wavelength you measured in your experiment, and you should get the speed of light or something pretty close to it. Here’s how it would go with the numbers I just gave: The frequency is 2.45 billion cycles per second. The wavelength is 12 centimeters (per cycle). Multiplying them together gives 29.4 billion centimeters per second. That’s pretty close to the accepted value for the speed of light, 30 billion centimeters per second.

the first microwave ovens were called radar ranges. They were not a commercial hit until the late 1960s. The first microwave ovens were too big, almost six feet tall, and extremely expensive, costing the equivalent of tens of thousands of dollars in today’s money.