Infinite Powers: How Calculus Reveals the Secrets of the Universe

by Steven H. Strogatz

it’s a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations.

Such equations describe the difference between something right now and the same thing an instant later or between something right here and the same thing infinitesimally close by. The details differ depending on what part of nature we’re talking about, but the structure of the laws is always the same.

“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.”

According to legend, Pythagoras felt it around 550 BCE when he and his disciples discovered that music was governed by the ratios of whole numbers. For instance, imagine plucking a guitar string. As the string vibrates, it emits a certain

note. Now put your finger on a fret exactly halfway up the string and pluck it again. The vibrating part of the string is now half as long as it used to be—a ratio of 1 to 2—and it sounds precisely an octave higher than the original note (the musical distance from one do to the next in the do-re-mi-fa-sol-la-ti-do scale). If instead the vibrating string is ⅔ of its original length, the note it makes goes up by a fifth (the interval from do to sol; think of the first two notes of the Star Wars theme). And if the vibratin...

calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the

original whole. This half of the subject is called integral calculus.

great advances in calculus grew out of the quest to solve the mystery of motion. The Infinity Principle came to the rescue, just as it had for curves. This time the act of wishful fantasy was to pretend that motion at a changing speed was made up of infinitely many, infinitesimally brief motions at a constant speed.

Albert Einstein applied calculus to a simple model of atomic transitions to predict a remarkable effect called stimulated emission (which is what the s and e stand for in laser, an acronym for light amplification by stimulated emission of radiation). He theorized that under certain circumstances, light passing through matter could stimulate the production of more light at the same wavelength

and moving in the same direction, creating a cascade of light through a kind of chain reaction that would result in an intense, coherent beam. A few decades later, the prediction proved to be accurate. The first working lasers were built in the early 1960s.

With the new understanding that calculus helped provide, HIV infection has been transformed from a near-certain death sentence to a manageable chronic disease—at least for those with access to combination-drug therapy.

Calculus began as an outgrowth of geometry. Back around 250 BCE in ancient Greece, it was a hot little mathematical startup devoted to the mystery of curves. The ambitious plan of its devotees was to use infinity to build a bridge between the curved and the straight. The hope was that once that link was established, the methods and techniques of straight-line geometry could be shuttled across the bridge and brought to bear on the mystery of curves. With infinity’s

help, all the old problems could be solved.

In 1899, the father of quantum theory, a German physicist named Max Planck, realized that there was one and only one way to combine these fundamental constants to produce a scale of length. That unique length, he concluded, was a natural yardstick for the universe. In his honor, it is now called the Planck length. It is given by the algebraic combination Planck length = √ ħG / c3 .

principle of buoyancy (a body immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced)

law of the lever (heavy objects placed on opposite sides of a lever will balance if and only if their weights are in inverse proportion to their distances from the fulcrum).

Mathematicians don’t

come up with the proofs first. First comes intuition. Rigor comes later. This essential role of intuition and imagination is often left out of high-school geometry courses, but it is essential to all creative mathematics.

What is the Method, and what is so personal, brilliant, and transgressive about it? The Method is mechanical; Archimedes finds the area of the parabolic segment by weighing it in his mind. He thinks of the curved parabolic region as a material object—I’m picturing it as a thin sheet of metal carefully trimmed into the desired parabolic shape—and then he places it at one end of an imaginary balance scale. Or, if you prefer, think of it as being seated at one end of an

imaginary seesaw. Next he figures out how to counterbalance it against a shape he already knows how to weigh: a triangle. From this he deduces the area of the original parabolic segment.

when two otherwise identical strings with

lengths in the ratio 3:2 are struck, they harmonize beautifully, separated in pitch by an interval known as a fifth. Similarly, strings in a 4:3 ratio produce a fourth. These numerical coincidences between harmony and geometry must have delighted Archimedes.

Archimedes’s legacy lives on today. Consider the computer-animated movies that our kids love to watch. 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. For example, here are three triangulations of a mannequin’s head. The more triangles we use and the smaller we make them, the better the approximation becomes.

atomic clocks are the key to pinpointing the location of anything on Earth to within a few meters. An atomic clock is a modern-day version of Galileo’s pendulum clock. Like its forebear, it keeps time by counting oscillations, but instead of tracking the movements of a pendulum bob swinging back and forth, an atomic clock counts the oscillations of cesium atoms as they switch back and forth between two of their energy states, something they do 9,192,631,770 times per second.

When you use the GPS in your phone or car, your device receives wireless signals from at least four of the twenty-four satellites in the global positioning system that are orbiting about twelve thousand miles overhead. Each satellite carries four atomic clocks that are synchronized to within a billionth of a second of one another. The various satellites visible to your receiver send it a continuous stream of signals, each of which is time-stamped to the nanosecond. That’s where the atomic clocks come in. Their tremendous temporal precision gets converted into the tremendous spatial precision we’ve come to expect from GPS.

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.

The global positioning system was developed by the US military during the Cold War. The original intent was to keep track of US submarines carrying nuclear missiles and give them precise estimates of their current locations so that if they needed to launch a nuclear strike, they could target their intercontinental ballistic missiles very accurately. Peacetime applications of GPS nowadays include precision farming, blind landings of airplanes in heavy fog, and enhanced 911 systems that automatically calculate the fastest routes for ambulances and fire trucks. But GPS is more than a location and guidance system. It allows time synchronization to within a hundred nanoseconds, which is useful for coordinating bank tra...

Differential calculus cuts complicated problems into infinitely many simpler pieces. Integral calculus puts the pieces back

together again to solve the original problem.

Although calculus culminated in Europe, its roots lie elsewhere. In particular, algebra came from Asia and the Middle East. Its name derives from the Arabic word al-jabr, meaning “restoration” or “the reunion of broken parts.”

geometry, as we’ve seen, was born in ancient Egypt; the founding father of Greek geometry, Thales, is said to have learned the subject there. And the greatest theorem of geometry, the Pythagorean theorem, did not originate with Pythagoras; it was known to the Babylonians for at least a thousand years before him, as evidenced by examples of it on Mesopotamian clay tablets from around 1800 BCE.

quadratic equations. These are equations in which, along with the usual constants (like 200) and linear terms (like x and y), the variables can also get squared or multiplied together, creating quadratic terms like x2, y2, and xy. (In Latin, quadratus means “square.”) Squared quantities had traditionally been interpreted as the areas of square regions. Thus, x2 meant the area of an x-by-x square.

The quadratic equations in x and y are the algebraic counterparts of the conic sections of the Greeks, the four kinds of curves obtained by slicing through a cone at different

angles. Here, in Fermat’s and Descartes’s new arena, classical curves were reappearing like ghosts from the mist.

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.

For any given choice of the four parameters, we calculate the discrepancy (in other words, the error) between the sine-wave fit and the actual data at every one of the twenty-seven points recorded throughout the year. A natural criterion for choosing the best fit is that the total error, summed over all twenty-seven points, should be as small as we can make it. But total error is not quite the right concept, because we don’t want the negative errors to cancel the positive ones and give the false impression that the fit has less error than it does. Undershoots are just as bad as overshoots, and both should be penalized; they shouldn’t be allowed to cancel out. For this reason, mathematicians square the errors at each point to

make the negative ones become positive. That way, they can’t possibly produce any spurious cancellations. (Here’s one place where the fact that a negative times a negative is a positive is useful in a practical setting. It makes the square of a negative error count as a positive discrepancy, as it should.) So the basic idea is to choose the four parameters in the sine wave in such a way that they minimize the total squared error of the fit to the data. Accordingly, this approach is called the method of least squares. It works best when the data follow a pattern, as they do here.

Sine waves are ideal for representing periodic patterns, but they are less well suited to representing sharply localized features, like the edge of a nostril or a beauty mark.

For this purpose, researchers in several different fields came up with a generalization of sine waves called wavelets. These little waves are more localized than sine waves. Instead of extending periodically out to infinity in both directions, they are sharply concentrated in time or space. Wavelets suddenly turn on, oscillate for a while, and then turn off.

When a ray of light passes from a thinner medium like air into a denser medium like water or glass, the ray bends toward

the perpendicular to the interface between the two media. When it passes from a denser medium into a thinner one, it bends away from the perpendicular,

The pivotal moment in the story of calculus occurred in the middle of the seventeenth century when the mysteries of curves, motion, and change collided on a two-dimensional grid, the xy plane of Fermat and Descartes.

when we multiply powers of ten, their exponents add, as 1 and 2 do here. The general rule is 10a × 10b = 10a+b.

related trend is that subtraction in the left column corresponds to division in the right column: 3 - 2 = 1 corresponds to 1000⁄100 = 10

a rule of thumb known to investors and bankers as the rule of 72. To estimate how long it will take to double your money at a given annual rate of return, divide 72 by the rate of return. Thus, money growing at a 6 percent annual rate doubles after about 72/6 = 12 years.

There are three central problems in calculus. They are shown schematically on the diagram below. The forward problem: Given a curve, find its slope everywhere. The backward problem: Given a curve’s slope everywhere, find the curve. The area problem: Given a curve, find the area under it.

when a point moves around a circle at a constant speed, its up-and-down motion traces a sine wave in time.