Editing 26: Fourier

{{comic
| number    = 26
| date      = November 2, 2005
| title     = Fourier
| before    = <big><big><span class="plainlinks">[https://web.archive.org/web/20060529063505/http://xkcd-drawings.livejournal.com/?skip=20#:~:text=Wednesday%27s%20Drawing%20%2D%20Fourier Original title</span>]: '''Wednesday's Drawing - Fourier'''</big></big>
| image     = fourier.jpg
| titletext = That cat has some serious periodic components
}}

This was the twenty-seventh comic originally posted to [[LiveJournal]]. The previous one was [[25: Barrel - Part 4]], and the next one was [[27: Meat Cereals]].

==Explanation==
A {{w|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.

Cueball is, in this particular comic, likely Jon from ''{{w|Garfield}}''. The name of Garfield's vet in the comic is Liz, and a recurring joke in that strip is Jon calling Liz to report various strange ailments befalling Garfield. "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 {{w|periodic function}}. In other words, the cat has repeating parts.

==Transcript==
:[Cueball is talking on the phone. A grotesque, spiky cat, who is apparently a Fourier transform, is looking at him.]
:Cueball: Hi, Dr. Elizabeth? Yeah, uh... I accidentally took the Fourier transform of my cat...
:Cat: Meow!

==Trivia==
{{incomplete|Someone who knows stuff please explain what they should've looked like.}}
*A user of this wiki, [[User:T7685|T7685]], created the graph of the reverse Fourier transform of the cat in this comic. Unfortunately, they don't have the shape of a normal cat. The resulting graphs are displayed below.
:[[File:26-1.png|500px|gallery|Absolute value of cat]][[File:26-2.png|500px|gallery|Components of cat]]

*After the publication of this comic, David Zaslavsky published a [https://www.ellipsix.net/blog/2013/09/interpreting-fourier-transforms.html fan version] on his blog [https://www.ellipsix.net/ ellipsix informatics], interpreting the transformed cat.

{{comic discussion}}

[[Category:Posted on LiveJournal| 27]]
[[Category:First day on xkcd.com]]
[[Category:Checkered paper]]
[[Category:Comics with color]]
[[Category:Comics featuring Cueball]]
[[Category:Math]]
[[Category:Cats]]
[[Category:Comics with lowercase text]]