Tools for thought: graphical algebra and Fourier analysis
Breath taking.<p>This technique could be used to visually prove many interesting properties of the continuous fourier transform.<p>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.)<p>What's interesting is that a visual proof like this can make a pretty non-trivial fact <i>visually obvious</i>.<p>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]<p>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.<p>[1] These ideas come from a conversation between Michael Nielsen and I, after we saw Steven's talk.