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.
This is amazing, no doubt. Not just the craft but the ideas and presentation.<p>However, I can't help but feel uncomfortable that this is not part of 'something bigger' -as many of the disconnected knowledge pieces across the web- that we could all contribute to (at the least for teaching/sharing/tweaking mathematics).<p>I am talking about a vague idea of what LaTeX was for unifying typesetting but for unifying collaborative proofs+visualization. Call it "VS" [1] to make the following analogy:<p>Before LaTeX, no one would have questioned your thesis for having handwritten equations on the gaps purposely left by you with the typewriter. It would not matter if you penciled the equation in or physically pasted a picture of it. No one would question the choice.<p>Similarly, before this hypothetical VS, people use Matlab, Mathematica, Sage, CoQ etc without anyone questioning any of the choices for a result/proof/visualization.<p>Even worse, before VS++ people publish papers on different journals with different degrees of access, many not even sharing the code used for arriving at such results, no way of verifying/visualizing unless you're running it in your head's OS.<p>[1] <a href="https://www.youtube.com/watch?v=KvMxLpce3Xw" rel="nofollow">https://www.youtube.com/watch?v=KvMxLpce3Xw</a> (Vulcan School)<p>[+] I may be repeating myself <a href="https://news.ycombinator.com/item?id=11129297" rel="nofollow">https://news.ycombinator.com/item?id=11129297</a>
This is beautiful.<p>Another nice explanation in a similar vein: <a href="http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/" rel="nofollow">http://betterexplained.com/articles/an-interactive-guide-to-...</a>
Spectacular.<p>The only reason I was ever good at Math is because I found relationships like these despite my math teachers' best attempts to crush exploratory math. I can still remember them chiding me for not doing math their way.<p>Certain great teachers aside, of course.
Agreed with other comments. Some really good insights. I wish I had those in school when I was learning this stuff:<p>Some gems:<p>> Every exponential curve of any base or offset can be drawn by scaling/ translating any other exponential curve, vertically/horizontally.<p>> ... a cosine (or sine) is essentially an approximate sequence of parabolas, alternating upside down. This tells you about the Taylor series expansion, and is not that surprising if you know that sine and cosine are solutions to a second order differential equation.<p>This one I found not completely intiuitive at first, had to think about it a bit:<p>> With complex numbers, it is trivial to twist the graph
Am I the only one who this doesn't work for? Tried in Chromium, Firefox and Conkeror; the best I get is a progress bar, then a set of axes with a single dot plotted on the y axis. :(
Impressive. No-brainer, awesome work.<p>I actually used the mathematics behind some algorithms to design my own Blog.<p>[0] <a href="http://leonardofed.io" rel="nofollow">http://leonardofed.io</a>
Cool<p>I wish the explanatory text weren't shoved off into a box below the cool stuff I want to be looking at. I think ideally the textual explanations should be overlaid on the graphical representations and at the very least they should be close to each other and within the same screen element. Personally I think I experience some type of distracting cognitive load when I have to switch contexts like that.
Very cool. I remember attending a lecture in Buenos Aires on "Fourier Analysis for Dummies" and this complicated mathematical term that had scared me for years suddenly seemed so straightforward. It's a neat thing to see others explain it in ways that don't involve all the equations.
Couldn't see it for lack of webgl but makes me wonder, how did people visualize/understand Fourier without all the neat abstractions, metaphors, and history ?