Music by Papa Diabate with the African instrument Kora at Polytechnique during the “semaine de la langues et cultures”. The first pic shows the special mop(?) that is used to clean it. The performer had invited students to come onto the stage and dance and there were indeed some students who accepted the offer. The best part about the music was the feeling of “lighting a fire in the night and contemplating on the stars in the olden days where there was no electricity”. It was blissful. I wish I could upload the music files that I had recorded but tumblr doesn’t allow .3ga files. Any idea how I could convert them to .mp3?
In the second animation, the transform is reapplied to the normalized sinc function, and we get our original rect function back.
It takes four iterations of the Fourier transform to get back to the original function. We say it is a 4-periodic automorphism.
However, in this particular example, and with this particular definition of the Fourier transform, the rect function and the sinc function are exact inverses of each other. Using other definitions would require four applications, as we would get a distorted rect and sinc function in the intermediate steps.
For simplicity, I opted for this so I don’t have very tall and very wide intermediate functions, or the need for a very long animation. It doesn’t really work visually, and the details can be easily extrapolated once the main idea gets across, I think.
In this example, it also happens that there are no imaginary/sine components, so you’re looking at the real/cosine components only.
Shown at left, overlaid on the red time domain curve, you’ll notice a changing yellow curve. This is the approximation using the components extracted from the frequency domain “found” so far (the blue cosines sweeping the surface). The approximation is calculated by adding all the components, by integrating along the entire surface (this is continuous, remember?)
As we add more and more of the components, the approximation improves. In some special cases, it is exact. For the rect function, it isn’t, and you get some wavy artifacts in some places (the sudden jumps, aka discontinuities). These are due to Gibbs phenomenon, and are the main cause of ringing artifacts. As you’ll probably notice, the approximation is pretty much dead on for the sinc function, as shown in the second animation.
The illustration shows the domains in the interval [-5,5], but the Fourier transform extends infinitely to all directions, of course.
The surface illustrated here isn’t too far off from the approach used in Fourier operators. If you consider the surfaces defined by z = cos(xy) and z = sin(xy), you get the cosine and sine Fourier operators. Having complex values lets you mix both into one thing.
The surface you see in the first animation is just z = cos(2πxy)sinc(πy). The Fourier transform can be thought of as multiplying a function by these continuous operators, and integrating the result. This can be very neatly performed using matrix multiplication in the discrete cases. (New drinking game: take a shot every time linear algebra shows up in any mathematical discussion.)
This also explains why the Fourier transform is cyclic after 4 iterations: rotating 90° four times gets you back to your original position. By using different rotation angles, you get fractional Fourier transforms. Awesome stuff.
Sadly, I had to reduce the images to 400 pixels wide instead of 500. Tumblr wouldn’t accept it otherwise. However, a HD video is also available:
This animation would probably look better with a different way of rendering that surface. Sorry, I don’t have anything better available at the moment, but I’ll work on it. If I do come up with something, I’ll post an update.
A fantastic set of typographic posters aimed at technology users of all ages showcased through a unique medium; the artist’s grandmother ! Enjoy these tidbits of wisdom and be sure to click each individual image for viewing at a higher resolution.