If you've spent any time on the Internet, at some point you've heard the phrase "Fourier transform." It's it's a mathematical discovery that's baked into everything from MP3s to quantum physics. Once you understand it, the world is your oyster. Luckily, Aatish Bhatia has a terrific article in Nautilus that will explain it all to you.

Bhatia begins:

Nine years ago, I was sitting in a college math physics course and my professor spelt out an idea that kind of blew my mind. I think it isn't a stretch to say that this is one of the most widely applicable mathematical discoveries, with applications ranging from optics to quantum physics, radio astronomy, MP3 and JPEG compression, X-ray crystallography, voice recognition, and PET or MRI scans. This mathematical toolâ€”named the Fourier transform, after 18th-century French physicist and mathematician Joseph Fourierâ€”was even used by James Watson and Francis Crick to decode the double helix structure of DNA from the X-ray patterns produced by Rosalind Franklin . . .

So what was Fourier's discovery, and why is it useful? Imagine playing a note on a piano. When you press the piano key, a hammer strikes a string that vibrates to and fro at a certain fixed rate (440 times a second for the A note). As the string vibrates, the air molecules around it bounce to and fro, creating a wave of jiggling air molecules that we call sound. If you could watch the air carry out this periodic dance, you'd discover a smooth, undulating, endlessly repeating curve that's called a sinusoid, or a sine wave. (Clarification: In the example of the piano key, there will really be more than one sine wave produced. The richness of a real piano note comes from the many softer overtones that are produced in addition to the primary sine wave. A piano note can beapproximated as a sine wave, but a tuning fork is a more apt example of a sound that is well-approximated by a single sinusoid.)