26: Fourier

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Fourier
That cat has some serious periodic components
Title text: That cat has some serious periodic components

Explanation[edit]

Ambox notice.png This explanation may be incomplete or incorrect: Missing reverse transform, or commentary on whether it is meaningful to take it. [to my uneducated eye it looks like half of a real-valued graph, but is phase information missing?] Missing guesses as to possible meaning of the name Elizabeth.
If you can address this issue, please edit the page! Thanks.

A Fourier transform is a mathematical function transformation often used in physics and engineering.

The theory is that any line graph can be represented as the sum of a bunch of sine waves of different frequencies and amplitudes. (The most obvious application is in analyzing a sound recording in terms of the different frequencies of sounds used.) So, for any line graph, you can produce another graph of the frequencies and their amplitudes. This can be done by evaluating an integral based on the function, which is referred to as "taking the Fourier transform" of the function. The form of the integral that needs to be taken is actually shown in the third line of the comic 55: Useless.

Unfortunately, Cueball has applied this "transform" to his cat. Indeed, whatever he has done has literally transformed his cat into the shape of an amplitude line graph. Although the cat seems to be alive and largely unharmed, it is clearly not in its familiar shape, and it is not clear if this condition is permanent or not. Notably, the fact the cat is still alive relates to an important property of Fourier transformation: the information of the original graph is fully preserved, and can even be reversed to produce the original graph. How a reverse Fourier transformation would apply to a transformed cat has yet to be seen.

"Periodic components" in the title text refers to the spikes in the graph. Because sine waves repeat themselves as you go along, the presence of large amounts of one particular sine wave in the Fourier transform graph (each spike) shows that the overall result (the initial graph) is likely to have parts that also repeat themselves, like a periodic function. In other words, the cat has repeating parts.

Transcript[edit]

[Cueball talks on phone. A grotesque-looking cat with many sharp vertical points looks on.]
Cueball: Hi, Dr. Elizabeth? Yeah, uh ... I accidentally took the Fourier transform of my cat...
Cat: Meow!

Trivia[edit]

  • This was the 27th comic originally posted to LiveJournal.
  • Original title: "Wednesday's Drawing - Fourier"
  • There were no original Randall quote for this comic.
  • This comic was posted on xkcd when the web site opened on Sunday the 1st of January 2006.
    • It was posted along with all 41 comics posted before that on LiveJournal as well as a few others.
    • The latter explaining why the numbers of these 41 LiveJournal comics ranges from 1-44.
  • One of the original drawings drawn on checkered paper.


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Discussion

Isn't the cat also imaginary because its Fourier transform isn't symmetric?

I feel like there's another joke in that his cat is "imaginary" or has complex components.

Shdwdrgn (talk) 06:33, 8 October 2014 (UTC)shdwdrgn

Picking up on shdwdrgn's comment above, how interesting would the Fourier transform of Schroedingers's cat be. I guess it would consist of two overlaid graphs neither of which would be certain until you actually looked at it.EditorGonk (talk) 09:38, 20 July 2018 (UTC)


Might this also be a Garfield joke? Garfield's veterinarian is named Liz. Although Garfield, being roughly a three-dimensional ovoid, would probably end up with a much different looking Fourier transform than what is depicted here.

--199.27.130.246 21:26, 9 October 2014 (UTC)

I think the transform may be of the movements of various parts of the cat. Cats tend to move their ears and heads a lot, and other parts, less so. What tipped me off is the spike at the tip of the tail. Cats typically twitch the very tip of their tail in a rhythmic fashion. 108.162.216.192 21:52, 2 March 2015 (UTC)

Coincidentially, the Fourier transform of a cat was used in a 2003 paper on the so-called phase problem in protein crystallography (figure 3) to illustrate the relevance of phase and amplitude information. See http://journals.iucr.org/d/issues/2003/11/00/ba5050/index.html and http://journals.iucr.org/d/issues/2003/11/00/ba5050/ba5050fig3.html

Can someone do a reverse fourier transform on the cat's graph and post it here please? --162.158.79.37 18:12, 14 July 2020 (UTC)Bumpf