As we listen to a piece of music, our ears perform a calculation. The high-pitched flutter of the flute, the middle tones of the violin, and the low hum of the double bass fill the air with pressure waves of many different frequencies. When the combined sound wave descends through the ear canal and into the spiral-shaped cochlea, hairs of different lengths resonate to the different pitches, separating the messy signal into buckets of elemental sounds.
It took mathematicians until the 19th century to master this same calculation.
In the early 1800s, the French mathematician Jean-Baptiste Joseph Fourier discovered a way to take any function and decompose it into a set of fundamental waves, or frequencies. Add these constituent frequencies back together, and you’ll get your original function. The technique, today called the Fourier transform, allowed the mathematician — previously an ardent proponent of the French revolution — to spur a mathematical revolution as well.
Out of the Fourier transform grew an entire field of mathematics, called harmonic analysis, which studies the components of functions. Soon enough, mathematicians began to discover deep connections between harmonic analysis and other areas of math and physics, from number theory to differential equations to quantum mechanics. You can also find the Fourier transform at work in your computer, allowing you to compress files, enhance audio signals and more.
“It’s hard to overestimate the influence of Fourier analysis in math,” said Leslie Greengard of New York University and the Flatiron Institute. “It touches almost every field of math and physics and chemistry and everything else.”
Flames of Passion
Fourier was born in 1768 amid the chaos of prerevolutionary France. Orphaned at 10 years old, he was educated at a convent in his hometown of Auxerre. He spent the next decade conflicted about whether to dedicate his life to religion or to math, eventually abandoning his religious training and becoming a teacher. He also promoted revolutionary efforts in France until, during the Reign of Terror in 1794, the 26-year-old was arrested and imprisoned for expressing beliefs that were considered anti-revolutionary. He was slated for the guillotine.
After taking part in the French Revolution and one of Napoleon’s campaigns, Jean-Baptiste Joseph Fourier revolutionized mathematics with the discovery of what’s now known as the Fourier transform. Julien-Léopold Boilly, Public domain, via Wikimedia Commons
Before he could be executed, the Terror came to an end. And so, in 1795, he returned to teaching mathematics. A few years later, he was appointed as a scientific adviser to Napoleon Bonaparte and joined his army during the invasion of Egypt. It was there that Fourier, while also pursuing research into Egyptian antiquities, began the work that would lead him to develop his transform: He wanted to understand the mathematics of heat conduction. By the time he returned to France in 1801 — shortly before the French army was driven out of Egypt, the stolen Rosetta stone surrendered to the British — Fourier could think of nothing else.
If you heat one side of a metal rod, the heat will spread until the whole rod has the same temperature. Fourier argued that the distribution of heat through the rod could be written as a sum of simple waves. As the metal cools, these waves lose energy, causing them to smooth out and eventually disappear. The waves that oscillate more quickly — meaning they have more energy — decay first, followed eventually by the lower frequencies. It’s like a symphony that ends with each instrument fading to silence, from piccolos to tubas.
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