In 1822, while studying heat conduction in metal bars, the French mathematician Joseph Fourier made a claim so audacious that the Academy of Sciences refused to accept it for publication: any periodic function, however jagged or discontinuous, could be written as a sum of pure sine waves. The claim was technically wrong as stated, technically correct under the right conditions, and one of the most useful ideas in applied mathematics. Every signal we record, transmit, compress, or filter today is processed by some descendant of Fourier's insight.
Fourier decomposition reveals that the world has two natures — a time domain in which waves rise and fall, and a frequency domain in which they appear as static collections of pure tones. The two are equivalent representations of the same information; the Fourier transform converts between them. The technique works because sine waves are the eigenfunctions of the differentiation operator, which means many physical systems — heat, sound, light, quantum mechanics — decouple in the frequency domain in ways they don't in the time domain. The applications are everywhere: JPEG compression discards high-frequency components your eye can't see; MP3 does the same for sounds your ear can't hear; MRI reconstructs images from frequency-domain data; communication systems pack multiple signals into different frequency bands; speech recognition uses spectrograms (Fourier transforms over short windows); the periodic table itself is a kind of Fourier decomposition of atomic structure.
The Fast Fourier Transform — Cooley and Tukey, 1965 — is one of the most important algorithms ever invented; without it, the digital signal processing revolution of the late twentieth century would have been impossible. Modern wireless (5G, Wi-Fi), medical imaging, audio engineering, financial-time-series analysis, and most of physics-based simulation rely on it. The deep idea — every signal is a chord — has the rare property of being technically rigorous and metaphorically generative at the same time.