Tools for thought: graphical algebra and Fourier analysis
Breath taking.This technique could be used to visually prove many interesting properties of the continuous fourier transform.For example, you can visually prove that the fourier transform is its own inverse, up to a constant factor. It's a bit tricky to communicate such a visual proof in text, but here's a sketch: First, consider the fourier transform of the dirac delta function, which this visualization would reveal to be a wave. Apply the transform again, and it's visually clear that the wave destructively interferes and has a transform of 0, except for a single value. This reveals that, applying the Fourier transform twice gives you the original function, up to a constant inverse. (If you work through this carefully, you see the constant is negative.)What's interesting is that a visual proof like this can make a pretty non-trivial fact visually obvious.You could take this further. I'm pretty certain that one could get visual proofs that taking the derivative is multiplying by i times the frequency in frequency space. Or get a visual proof of the convolution theorem. If you have a visual proof of the convolution theorem, you can probably get a visual proof of the central limit theorem. [1]I actually tried to illustrate this visual proof of the Fourier transform's inverse several years ago. I tried to make static diagrams, not an animated version. I failed. I say this to emphasize how impressive Steven's feat is: not only did he make a beautiful visualization of this, not only did he make it a live animation, but he did it in a freaking web browser. It's really incredible.[1] These ideas come from a conversation between Michael Nielsen and I, after we saw Steven's talk.