In my last post [^], we saw how it is possible for people to argue that the instantaneous action-at-a-distance (IAD) is not a confusion of the Fourier theory—it is just a feature.
I also said (with a minor editing):
“Knowing for a fact that we have these two classes of theories, which, while speaking about the same mathematical equation which has been studied for a couple of centuries, have completely different things to say when it comes to an important issue like IAD—that is important.”
And I had asked you to think of a reason why mathematical physicists, mathematicians and physicists might not have “caught” a matter this “simple.” I had asked if you had any clue:
“…what could possibly the reason as to why a matter this “simple” has not been “caught” or “got” or highlighted by any single mathematician so far—or a mathematical physicist, for that matter? Why do people adopt one mind-set, capable of denying IAD, when in the stochastic realm, and immediately later on, adopt another mind-set that explicitly admits IAD? Why? Any clue?”
So, have you got any clue at least by now?
To make life a little better for you as far as this Fourier theory and all is concerned, let me point out to you a really cool collection of Web pages on the Fourier theory.
I was almost tempted to lift the entire material from there and repost it here (with proper acknowledgements, of course). But on the second thoughts, I have decided to just cache those Web pages to my hard disk for the time being [just in case!], and instead send you to the original URL.
The Web pages in question start here: Dr. Kevin Cowtan’s Picture Book of Fourier Transforms [^].
Make sure to see all the pages, right from “Introduction” through “The Gallery.” Also see the “Other topics” mentioned on the starting page.
My favorite page? It’s the “Animal Magic,”i.e. mixing two images by first taking their FFTs, then taking the phases for one image and magnitudes for the other, and then taking a reverse FFT of this mix-up. I liked that idea (which was completely new to me). And, the rather unintuitive conclusion we reach: “the image which contributed the phases is still visible, whereas the image which contributed the magnitudes has gone!” [^].
And, also check out Dr. Cowtan’s Web page on the convolution theorem [^]. This is the shortest—and separately, the best—explanation on the convolution theorem that I have ever run into, anywhere, including in the books and popular expositions. Yes, dear reader, the operation illustrated on this page is the same as what the Wiki [^] and the Wolfram site [^] say about it.
I now assume that you have explored that Web site, and have returned, and are wondering what more I have to say about it all.
Well, this whole thing (Dr. Cowtan’s site and what I write below) is a bit of a diversion, really speaking. Its purpose is not to clue you in on the clue which I have in mind concerning the issue which I mentioned in my past post. This post is related to the Fourier theory, but not directly with that problem.
Anyway, if you would like to explore a bit further, and are inclined towards programming, I suggest an exercise:
Consider the XI-standard physics problem of throwing a ball up in the frictionless air and catching it after it has followed the parabolic path back down to the ground. Write a GUI-based program that shows the movement of a small ball of finite size in tracing the parabolic trajectory. (You don’t have to numerically solve Newton’s equations; you may assume that the closed-form end-solution, viz. the equation of the parabola, is known already.) Consider using OpenGL and using two threads so that the program doesn’t hog your computer’s resources or freeze it. We will need interactivity especially for the next part.
Now, for the second part: Suitably discretize the parabolic path of the ball, and show in a separate window, in real time, the evolution in the instantaneous 2D FFTs of the ball as it traverses its path through the physical space.
In case you aren’t inclined towards programming, but still would like to explore the above exercise a bit with a ready-made tool (having a lot of limitations: no automatic updates, only 1D, coarse discretization, etc.), check out the “FFT Laboratory” page maintained by Dave Hale at a Stanford server [^]. (I am not sure if he works at Stanford. BTW, he also has a couple of interesting papers on meshing [^].)
Now take a moment to appreciate what lessons we can draw out of this whole exercise.
First, realize that both parts of the simulation—the one in the physical space and the other one in the frequency space—are, strictly speaking, what physicists call models. Now, think: Which one looks like a real description of the moving ball to you—the one in the frequency space, or the one in the physical space? Why?
Second, what opinion can you possibly develop on those physicists who assert that a frequency-space-based description is necessary at the most fundamental level in a theory of physics?
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And, don’t go around so totally clueless on that issue of my last post—why mathematicians could not have got such a “simple” thing.
Think about it. (LOL!)
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A Song I Like:
[BTW, I you know that I have been jobless throughout 2012. I have been kept out of the CAE jobs, possibly deliberately, and very possibly due to that hassle and tussle which I have been having with those JPBTIs, Americans, etc. Realizing the deliberateness of it all, and once the academic jobs, too, were deliberately denied to me, I then have interviewed with an IT company. While I still go without a job, I think they are considering seriously to offer me a job. [BTBTW, noting this fact publicly is my way of pressurizing them to give me that job ASAP.] Conclusion? The inclusion here of this section is still very much on the whim.]
(Hindi) “meri zindagi mein aaye ho, aur aise aaye ho tum…”
Singers: Sunidhi Chauhan and Sonu Nigam
Lyrics: Javed Akhtar
[As usual, another edit may be due.]