The raw infrasound data Garces collects requires a great deal of signal processing and analysis before real-time surf infrasonic monitoring can yield useful insights into ocean wavefronts. Because the waves that eventually hit the shores of Kona are an accumulation of many different waves of varying frequency, part of that processing involves breaking down complex waveforms into the individual component waves. This can be done thanks to a method devised by eighteenth-century French mathematician Jean-Baptiste-Joseph Fourier, a procedure known as a Fourier transform. I wasn’t able to find any historical evidence that Fourier, like Twain, ever tried his hand at surfing. But I’m sure Fourier would have made an excellent surfer—at least in theory. This was a man well versed in periodic functions.
Fourier had a gift for making waves. Born in 1768, he was the son of a very fertile tailor in the village of Auxerre; Fourier had eleven siblings, as well as three half-siblings from his father’s first marriage. Orphaned by age ten, the young Jean-Baptiste received an early rudimentary education at a local convent, thanks to a recommendation by the Bishop of Auxerre, and he proved such an apt pupil that he went on to study at the École Royale Militaire of Auxerre. There he fell in love with mathematics, although he initially planned to enter the priesthood. Math won out in the end; by 1790 Fourier was teaching at his alma mater in Auxerre.
Perhaps his desire to focus on mathematical research—and his inability to accomplish much of significance in his earliest years—was influenced by the tumultuous times in which he lived. Revolution was brewing in France. Fourier was sympathetic at first to the revolutionary cause, drawn by “the natural ideas of equality,” and a hope “of establishing among us a free government exempt from kings and priests.” He joined his local Revolutionary Committee but soon regretted it, as the ultraviolent Reign of Terror gripped France and thousands of nobles and intellectuals fell victim to the guillotine. The streets of Paris literally ran with blood.
It was frighteningly easy to run afoul of the murderous mob mentality that prevailed during the Terror; the movement soon splintered into squabbling factions, despite sharing similar goals, and rampant hysteria spread throughout France. Wise men kept their heads down and tried not to attract attention, as almost anyone could be accused of treason for the slightest perceived infraction against the new republic in that volatile environment. Fourier made the mistake of defending the stance of his own Auxerre faction before a rival sect while on a trip to Orléans. In July 1794, he was arrested and imprisoned for the views he’d expressed on that trip, and found himself facing the guillotine.
He was fortunate that his imprisonment occurred just before Maximilien Robespierre—mastermind of the Reign of Terror—ran afoul of his own revolution and lost his head to the angry mob he helped incite. With the death of Robespierre, the Revolution lost steam, and Fourier and his fellow prisoners were freed. Fourier had the good fortune to be selected for a new teacher-training school to help rebuild France, where he studied under three of the most prominent French mathematicians: Lagrange, Laplace (who wisely fled Paris during the Terror), and Gaspard Monge. By September 1795, Fourier was teaching at the prestigious École Polytechnique.
All this occurred before Fourier turned thirty. But the quiet life of contemplation still eluded him. A few years after his academic appointment, he joined Napoléon’s army as a scientific advisor when Napoléon invaded Egypt. Mostly he engaged in archaeological expeditions and helped found the educational Cairo Institute, as Napoléon’s military fortunes in Egypt waxed and waned. By 1801, Fourier was back in France, teaching, until Napoléon whimsically appointed him prefect in Grenoble. At long last, Fourier was in a stable, peaceful environment where he could focus on mathematics—and he promptly stirred up a mathematical controversy.
MIXING AND MATCHING
The culprit was a single equation describing how heat traveled through certain materials as a wave. Fourier concluded that every wavelike “signal,” no matter how complex, could be rebuilt from scratch by adding together many different waves mixed together according to a specific “recipe.” In other words, complicated periodic functions can be written as the sum of simple waves mathematically represented by sines and cosines (now known as the Fourier series). We can figure out which waves are present in a complex signal by taking an integral over all possible waves. That is the Fourier transform.
Fourier transforms are difficult for a beginning calculus student to grasp, and more complex signals require powerful computers to crunch the numbers, but the overall concept is straightforward enough. You just take apart the original signal to determine the “ingredients,” and then figure out how to rebuild that signal with a mixture of the same component sinusoid waves.
It’s a bit like trying to re-create at home your favorite restaurant’s spécialité de la maison, except you have to guess at the ingredients. The more sinusoids we use, the more accurately the resulting reconstructed waveform resembles the original—much as estimating the area underneath a curve gives a more accurate result if you use more and more rectangles in the method of exhaustion. Anytime we are adding together many different smaller pieces that add up over time, we are taking an integral.
There is a neat trick to determining whether a given waveform is an ingredient in our original signal. Earlier we saw two simple sine waveforms, representing the functions sin(x) and sin(2x). If we multiply sin(x) by itself, we get a wave that looks like this:
Note that it oscillates entirely above the x axis, unlike the original sine wave, which oscillated equally above and below the x axis. If we integrated it, the total area would gradually accumulate; it would just go up and up, with no subtractions. This is how we know that sin(x) is a component of our original signal—indeed, it is the only component wave of our original signal. In contrast, if we multiply sin(x) by sin(2x), we get a resulting wave that looks like the graph at the top of page 244.
This time, it oscillates fairly equally above and below the x axis. If we integrated it, the total area would oscillate around 0, because sometimes the area adds to the total, and sometimes it subtracts. This tells us that sin(2x) is not a component of our original signal. We would get a similar result if we multiplied sin(x) by sin(1.1x), sin(3x), or any other wave, because our original signal was not a complex waveform, but consisted of one simple wave as the sole ingredient.
Let’s see what happens when we have a signal that adds two waves together: the function sin(x) + sin(2x), which looks like this:
Now we perform the exact same process for each possible sine wave that could be a component. For instance, multiply the above wave by sin(x), and we get this:
Since most of the oscillation occurs above the x axis, we know that if we integrated it, the total area would accumulate, indicating that sin(x) is one of the components of our original signal. But if we multiply our original signal by sin(3x), we get something that looks like the illustration on the illustration below.
This tells us that sin(3x) is not a component of our original signal, because it oscillates equally above and below the x axis. We would get a similar result for every other possible wave we tried—a variable that is commonly designated in an equation by the Greek letter omega (ω)—that was not either sin(x) or sin(2x), because those are the only two ingredients of our original waveform. Ultimately, what the Fourier transform does is turn our original function of x into a function of ω. The integral is what makes that transformation complete.
Digital signal processing (DSP) would not be possible without Fourier analysis. DSPs are microelectronic devices that determine which sound wave is required to cancel noise. A DSP contains a resonator that vibrates in response to specific incoming frequencies. It then re-creates that sound wave—minus the frequency it is trying to cancel—and amplifies it through speakers or headphones. The end result is near silence. Most cell phones, CD players, and hearing aids now contain one or more DSP devices. The Fourier transform is a mathematical resonator, an efficient tool to filter signals.
A DSP first selects a sampling of a given signal measured at regular intervals. Let’s say we have 1,000 such samples of that signal, perhaps the infrasonic murmurings of a breaking ocean wave. We don’t know exactly how many “partial” sine waves make up that complex waveform, but the number of samples gives us an upper limit and a lower limit. Once we have that range of values, we can determine how many sine waves would fit between those two limits. In most cases, we will need as many sine waves as we have samples (i.e., 1,000) to perfectly reconstruct a given signal.
So now we know the frequency of our component sine waves. We also need to know their amplitude, that is, how much of each partial sine wave to mix in as we rebuild our original signal. If a given signal is heavy on the bass, there are more low-frequency sine waves mixed in than high-frequency ones, and vice versa if the signal is shrill with more treble in the mix. How can we figure that out? Why, by taking an integral, of course. We multiply that sine wave with signal samples and add the results together to get the amplitude of the partial sine wave at the frequency of interest. We do this for every possible frequency of sine wave, weed out those that we identify as ingredients, and voilà! We end up with the recipe for rebuilding that signal.
That evening finds me unwinding from the day’s exertions over a refreshing tropical cocktail in the hotel’s outdoor bar. The sun is setting, casting a pinkish orange glow over the water as hotel staff readies for the night’s luau performance. I find myself listening to the rhythmic cycle of waves crashing on the shore and reflecting on the fact that I can hear those waves because my brain is performing Fourier transforms constantly. It’s an essential part of how we hear.
The brain senses the incoming pressure wave and performs a Fourier transform on that signal to identify the frequency and amplitude of the “sound.” The ear measures change in pressure as a function of time. Sometimes it is a single note; sometimes it is several notes together, as in a musical chord; in each case, the brain uses the Fourier transform to determine which components make up the total sound wave. Similarly, every time we gaze at a sunset and identify specific hues—a spectacular orange-red, or a more subtle pinkish glow—our brain has taken a Fourier transform to isolate specific frequencies of light. And when a surfer eyeballs incoming ocean waves, his or her brain is making similar calculations.
The Fourier transform has a personal significance as well. Shortly after becoming engaged, Sean and I drove from a conference in San Francisco to our new home in Los Angeles via the scenic route along the Pacific Coast Highway. At sunset, we stopped briefly to refuel just north of Malibu and found ourselves admiring the brilliant orange, red, and purple hues stretching across the darkening horizon, savoring the peaceful sound of waves lapping against the shore. It was the perfect romantic setting to cap off a long day’s drive. Sean is nothing if not romantic. He is also the quintessential physicist. So he put his arms around me and whispered, “Wouldn’t it be fascinating to take a Fourier transform of those waves?”
I will never listen to ocean waves or view a beautiful sunset in quite the same way again. That is perhaps the greatest gift one can gain by delving into calculus: It is a whole new way of looking at the world, accessible only through the realm of mathematics. I looked out over the ocean that evening and saw a picture-perfect ocean sunset, but there was so much more that I missed. Sean looked out onto the same scene and saw the rich complexity of nature expressed in mathematical symbols, the fundamental abstract order lying just beneath the surface.
EPILOGUE
The Mimetics of Math
I’m very good at integral and differential calculus,
I know the scientific names of beings animalculous;
In short, in matters vegetable, animal, and mineral,
I am the very model of the modern Major General.
—GILBERT AND SULLIVAN,
The Pirates of Penzance
A pretty peach-hued building with an octagonal turret facing the Pacific Ocean is nestled on the edge of the University of California campus in Santa Barbara. This is the Kavli Institute of Theoretical Physics, where the world’s best physicists gather to exchange ideas that will usher in the revolutionary breakthroughs of tomorrow. The setting is idyllic, right next to the beach, so the siren song of sun and surf inevitably vies for my attention. On this particular day, the science is winning. There is a “blackboard lunch talk” by Joe Burns, a friendly and engaging astrophysicist from Cornell University. He is among the many scientists involved with analyzing data collected by NASA’s Cassini spacecraft orbiting Saturn to learn more about this distant planet—especially its mysterious rings.
It has been my custom during technical talks at KITP to focus on the concepts and let my eyes glaze over whenever an equation appears; much of the math is far too advanced for a fledgling calculus student to follow anyway. At first, Burns’s talk—while less technical than some of the brain-melting lectures I’ve attended—looks to be no exception. Inevitably, Burns turns to the blackboard and starts scratching out equations. But this time, I recognize the notation. Burns is taking a derivative. In a flash, I realize this means he is calculating a varying rate of change: the minute changes in velocity of the millions of icy particles (ranging from the size of seashells to surfboards) that orbit Saturn and make up its rings.
That lecture was a “mimetic moment” for me—the point where the abstract symbols in my calculus books finally began to make some sense, because I could connect them with something recognizable in the real world. In ancient Greece, mimesis referred to the artistic representation of nature, although two philosophers differed dramatically in their interpretations of the term. In one corner, I give you Plato, of cave-allegory fame, who believed in a divine realm of Ideal Forms. All creation, including Nature, was imitation in his eyes, and artistic imitation was by definition twice-removed from the Ideal. Ergo, all art (created fictions) is inferior to the “real” world, which is in turn inferior to the realm of Ideal Forms.
In the other corner, we have Aristotle, who took some time off from speculating that we see by shooting rays of light out of our eyes that reflect off nearby objects, to write his famed treatise Poetics. Aristotle was more forgiving of mimetic make-believe, for he thought that human beings have an inherent need to create artistic fictions as a form of catharsis, although he valued tragedy over comedy, via a rather convoluted process of reasoning. (He was wrong about how human vision works, too.) Our modern aesthetic still owes something to Plato and Aristotle, both of whom distinguished between diegesis, the act of telling, such as indirect narration of action or lecturing to students about calculus, and mimesis, the act of showing a character’s internal thoughts and emotions via external actions. It’s a dictum of modern entertainment: Show, don’t tell.
Anyone who’s taken Philosophy 101 could tell you that much. But in 1946, a literary scholar named Erich Auerbach adapted the concept of mimesis in what his biography at Lerhaus. org claims is “one of the most ambitious works of literary theory ever undertaken.” Mimesis: The Representation of Reality in Western Literature is pretty much required reading for serious students of literature; it had a profound effect on my undergraduate self, and a copy still graces my bookshelves. Auerbach analyzes literary conventions throughout the history of Western Europe and how they create “a lifelike illusion of some ‘real’ world outside the text.” My college English professor described the mimetic moment as the point at which one makes the critical connection between one’s own experiences and the artistic work and realizes, “Aha! This is that!” This kind of emotional and intellectual resonance on the part of the audience is what makes the creative arts so powerful.
I’ve spoken to many a scientist who was inclined to agree with Plato in devaluing fiction, which is a shame, because I would argue that created fictions present a uniquely effective teaching tool, a way to supplement rather dry college lectures (diegesis) with a dose of creativity (mimesis) to spark students’ excitement and interest.53 Show; don’t just t
ell. The mimetic moment is a critical component of acquiring true knowledge—actual learning, as opposed to memorizing facts by rote. Learning science, math, or any other subject is all about making that critical connection.
While I was at KITP, I got to know mathematician Bisi Agboola, who teaches at UCSB. Bisi was educated in the United Kingdom and failed most of his math classes through their equivalent of high school: “I found it dull, confusing, and difficult.” As a child, he was determined to find a career in which he wouldn’t need any math, finally announcing to his skeptical parents that he would be a woodcutter. He was crushed when they pointed out that he would need to measure the wood.
But one summer he encountered a Time-Life book—simply titled Mathematics, by David Bergamini—on the history of mathematics, from the Babylonians up until the 1960s. “It captured my imagination and made the subject come alive to me for the very first time,” he said, and it changed his mind about this seemingly dry subject. He realized there was beauty in it, and he wound up teaching himself calculus. Today he is a mathematician specializing in number theory and exotic multidimensional topologies. But he still doesn’t much like basic arithmetic: “I find it boring.”
The Calculus Diaries: How Math Can Help You Lose Weight, Win in Vegas, and Survive a Zombie Apocalypse Page 21
