The continuous Fourier transform takes an input function f(x) in the time domain and turns it into a new function, ƒ̂(x) in the frequency domain. (These can represent other things too, but that’s besides the point.)
(Tumblr kept rejecting the proper sized GIFs, so they may look blurry, pixelated or compressed to you. There’s also HD video.)
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.
NOTE: This animation is a follow-up to the previous one on time/frequency domains, showing discrete frequency components. Check that one out too, as it may help with understanding this one.
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.
The universal language of mathematics, expressed in different forms. Though I would have liked to include illustrations from other cultures: Mayan, Indian, Egyptian and others.
A Globe Girdling Theorem
The Pythagorean Theorem, first expounded more than 2,000 years ago, was familiar all over the civilized world by 17th century. At top left is a Greek text of Euclid’s proof, and with it its five translations. Although the Chinese text is only 350 years old, the Chinese were actually familiar with the theorem at about the time of Pythagoras.
What Are Little Planet Projections?
We’ve seen these wonderful forms of photographic manipulation many times before on APOD, but how are they done?
To generate these images we start with a spherical (equirectangular) panorama. This is an image where the x-axis corresponds to the longitude around a sphere (0-360 degrees) and the y-axis is the latitude (-90 to 90 degrees).
For any longitude or latitude position on a sphere we can retrieve the colour directly from the corresponding x,y coordinates on the panorama image. A proper equirectangular panorama should be twice as wide as tall, e.g. 1024x512 pixels.
Stereographic projection is a mapping that projects a sphere onto a plane, as illustrated with the world map below. It is conformal, which means that it preserves angles locally (note the grid lines still cross each other at right angles) although it doesn’t preserve areas or distances.
As we already have the colour of each longitude and latitude point on a sphere from the equirectangular panorama the inverse stereographic projection formulas are used, as described by Mathworld.
Want to give it a try? A few folks have already written tutorials on this emerging art form. You can check out a couple of these tutorials over at flickr here and here, but I’m sure there are many more on the web. Now go on and make a couple of little planets of your own!
Want to submit sky or astro photography to CWL? Head over to the submissions section. If you’ve got some of your own or generally favorite awesome photos you’d like to share with us.. Don’t be shy! If it’s awesome, best believe it will be posted.
Geometric Gifs by Matthew DiVito
Using nothing but soap and a macro lens, Janet Waters photographs mesmerizing patterns on colored backdrops.
But she hasn’t stopped there, she’s using her Flickr to create a “visual library” for all of her University students. Packed with experimental photo projects galore, her stream is well worth a look.
Melba Roy, NASA Mathmetician, at the Goddard Space Flight Center in Maryland in 1964. Ms. Roy, a 1950 graduate of Howard University, led a group of NASA mathmeticians known as “computers” who tracked the Echo satellites. The first time I shared Ms. Roy on VBG, my friend Chanda Prescod-Weinstein, a former postdoc in astrophysics at NASA, helpfully explained what Ms. Roy did in the comment section. I am sharing Chanda’s comment again here: “By the way, since I am a physicist, I might as well explain a little bit about what she did: when we launch satellites into orbit, there are a lot of things to keep track of. We have to ensure that gravitational pull from other bodies, such as other satellites, the moon, etc. don’t perturb and destabilize the orbit. These are extremely hard calculations to do even today, even with a machine-computer. So, what she did was extremely intense, difficult work. The goal of the work, in addition to ensuring satellites remained in a stable orbit, was to know where everything was at all times. So they had to be able to calculate with a high level of accuracy. Anyway, that’s the story behind orbital element timetables”. Photo: NASA/Corbis.
World’s largest prime number, all 17 million digits, discovered
FoxNews.com: Curtis Cooper at the University of Central Missouri in Warrensburg has found the largest prime number of all at this time.
As part of the Great Internet Mersenne Prime Search (GIMPS), a computer program that networks PCs worldwide to collectively hunt for a special type of prime number, Cooper discovered a prime number with 17,425,170 digits.
The number is the first prime discovered in four years.
[Images: National Center for Education Statistics]