I have always been fascinated by complex concepts explained simply - and so simply that I can start using the concept to see things around me in “shorthand”.

I saw this explanation of the Fourier Transform by Aatish Bhatia in 2013 in his blog post and have never forgotten it. Here are a couple of excerpts from it that stood out.

“.. The Fourier transform is like a mathematical prism—you feed in a wave and it spits out the ingredients of that wave—the notes (or sine waves) that when added together will reconstruct the wave… The Fourier Transform is a recipe—it tells you exactly how much of each note you need to mix together to reconstruct the original wave..”

“.. Here’s why Fourier’s trick is useful. Imagine you were talking to your friend over the phone and you wanted to get them to draw (a) squarish wave. The tedious way to do this would be to read out a long list of numbers that represent the height of the wave at every instant in time. With all these numbers, your friend could patiently stitch together the original wave. This is essentially how old audio formats like WAV files worked. But if your friend knew Fourier’s trick, you could do something pretty slick: You could just tell them a handful of numbers—the sizes of the different (sine waves and they could reconstruct the original wave)…”