# Toolbox – The Fourier Transform

Things that repeat themselves abound in nature. The back-and-forth motion of a branch in the breeze, the repeating hexagons of a honeycomb, and the popularity of the 80s, are just a few examples. Despite living in a world of oscillations, for most of history people have lacked the tools to simply describe repetitive things. As early as 700 B.C., Babylonian astronomers were studying the movements of the Sun-Moon- Earth trio, a system rich in oscillations. They tracked numerous celestial events including the interval between moonset after a new moon and sunrise of the next day, shown as the jaggy black line in the figure.

The Babylonian data appears erratic because the subject of observation (the Moon) rotates around the observation platform (Earth), which itself rotates and orbits the Sun. Each of these oscillations expresses itself in the data, along with many other more subtle celestial motions. Wouldn’t it be nice if we could take this complex oscillation and uncover the simple oscillations that comprise it, like the Sun’s period? The bottom half of the figure shows just that; we break down the Babylonian data, a complex oscillation, into the simplest oscillators, sinusoids, with the largest peaks showing which oscillation periods are expressed most strongly in the signal. From this view, it is clear that the signal is dominated by two components—the Sun’s familiar 12-month period, elusive in the original data, shows up as the largest peak and a prominent peak around 14 months signals the presence of another important astronomical period, the full moon cycle. We call this useful view the frequency domain representation, since our contributors are sinusoids of different frequencies. Importantly, the time and frequency domain representations are just different views of the same data, and as the upward arrow on the right indicates, we can move freely between the two.

The bridge between the domains is the Fourier Transform, but it wouldn’t be formulated until many centuries after the Babylonians. Eighteenth and 19th century brainiacs like Gauss and Bernoulli made big steps toward understanding how a signal could be deconstructed, but their solutions were not general enough. A one-size-fits-all method for most any imaginable signal was nowhere to be found. Many leading mathematicians doubted if such a thing existed.

Enter Jean Baptiste Joseph Fourier, a scientist-mathematician. Fourier followed a path not all that different from many of today’s highly motivated students. He took a prestigious internship in politics (as his town’s spokesperson during the French Revolution), attended university (to study under math legends Lagrange and Laplace), and even spent some time abroad doing international development work (as Napoleon’s scientific advisor, governor of Lower Egypt, and secretary of the Institut d’Egypte), although in those days, development looked a lot more like conquest.

After that, he settled down to formulate a complete mathematical description of heat transfer. Though he had the equation that describes how heat moves, solving it required performing precisely that dubious proposition of decomposing any signal into a sum of sinusoids. A tough break, indeed. But he kept at it long enough to devise an equation that did exactly that. This method is the core piece of math behind the Fourier Transform, the bridge between time and frequency domains.

Credit: Graphic: Asako Miyakawa Data: Lis Brack-Bernsen & Mattias Brack

The ability to hop in and out of the frequency domain revolutionized the analysis of oscillatory data. X-ray crystallographers study the structure of the molecules by coercing molecules into crystals, which are vast periodic structures. Geoffrey Feld, chemistry graduate student and crystallographer at UC Berkeley notes that “since we don’t have the ability to just take a picture of the atoms in their native state, we have to deduce it experimentally with math.” Whereas in the Babylonian data, the oscillations took place over time, like a swaying branch, crystals contain oscillations in space, like a honeycomb. Luckily, the math doesn’t mind either way, and we can leverage the same logic of Fourier’s method to understand the data.

As mere humans with mere human senses, we rarely occupy the ideal vantage point for answering the questions we ask. The Babylonians were too close to their object of interest, and crystallographers are too far from theirs. In these cases, we can augment our senses with tools for making measurements and tools for making sense of those measurements. In the same way telescopes gave us eyes to see far away and microscopes gave us eyes to see very small things, Fourier’s method provides us with a mathematical lens to see the oscillations that fill our world.