Editing 1381: Margin

{{comic
| number    = 1381
| date      = June 13, 2014
| title     = Margin
| image     = margin.png
| titletext = PROTIP: You can get around the Shannon-Hartley limit by setting your font size to 0.
}}

==Explanation==
{{incomplete|This explanation is a mess! To a layman, all four paragraphs make no sense. Even to someone who understands the math, its so messily written and assumes such a high level of comprehension that it goes over my head, and I understand the comic! Must be fixed!}}

This is a reference to {{w|Fermat's Last Theorem}}, of which {{w|Pierre de Fermat}} claimed he had a proof that was too large to fit in the margin of a copy of ''{{w|Arithmetica}}''. Despite its simple formulation, the problem remained unsolved for three centuries; it was cracked only with advanced techniques developed in the 20th century, leading many to believe that Fermat didn't actually possess {{w|Fermat's Last Theorem#Fermat's conjecture|a (correct) proof}} (see [[#trivia|trivia]]).

In the comic, the person writing in the margin attempts to pull a similar trick, without actually having any proof, by claiming that he has found a proof that information is infinitely compressible, but pretending not to be able to show it due to lack of space in the margin. In this particular case, however, this approach backfires, precisely because if information was actually infinitely compressible, the writer ''would'' be able to fit the proof in the margin (due to his own proof). The writer realizes that if he had a proof he should be able to fit it into the margin, and thus he realizes that he cannot pull this trick. Or perhaps the writer really thought he had a proof, but then realized that his statement was a counterexample, and was disappointed that his idea for a proof was wrong.

What it seems he did not realize, is that it would be impossible to read the proof if the writer actually was able to compress his proof to fit in the margin. This is because you would need to know the algorithm described in the proof before you could decompress the proof text so you can read it. So he could actually have used this trick instead, writing that he had compressed it into - say a dot "'''.'''" - and then people would have to find his proof to read it. And since they cannot find such a proof - they could not check his dot. Unfortunately this would also have backfired - because there is already a {{w|Pigeonhole principle#Uses and applications|proof that this is not possible}}!

Another thing that he probably didn't realize, is that finding a proof for something being possible does not necessarily mean inventing an actual algorithm to do that particular thing. If the person claimed having found a {{w|Existence theorem|non-constructive proof}} for such an algorithm, his statement at least wouldn't contradict itself.

The title text, yet another [[:Category:Protip|protip]], makes a reference to the {{w|Shannon–Hartley theorem}}, which limits the maximum rate at which information can be transmitted. Setting the font size of text only changes its ''representation'' on the screen, and not the actual characters themselves. Trying to decrease the amount of space needed to store or transmit it like advised would be nonsensical. Another possible interpretation is that if you set the font size to 0, the text cannot be seen, and therefore, nothing is being transmitted period.

In the case of actual printed paper, decreasing the font size is valid technique for information compression (more information on the same page), as used in ie. {{w|microform}}.  However, this comes at the cost of an increased spatial bandwidth (number of black/white transitions per distance). In the end, the resolution of the printer/paper/microscope chain limits the minimal font size that remains useable (above the {{w|Nyquist rate}}).

==Transcript==
:[Written on the right margin of a page:]
:I have
:discovered
:a truly
:marvelous
:proof that
:information
:is infinitely
:compressible,
:but this
:margin is too
:small to...

:...oh

:never mind :(

==Background to Fermat's Last Theorem==
{{incomplete|What in the world went on here? Did everyone who wrote this do it in their sleep? This is unbelievably messy! I know some of this story, and I've come out of reading this more confused than before!}}
*Fermat's Last Theorem states that no three positive integers ''a'', ''b'', and ''c'' can satisfy the equation ''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup> for any integer value of ''n'' greater than two.
**In the case with n=2, a b and c are the sides of a {{w|Pythagorean theorem|right triangle}}. There are an infinite number of integer solutions for a, b and c, such as ''3''<sup>''2''</sup> + ''4''<sup>''2''</sup> = ''5''<sup>''2''</sup>. This was known to Euclid, but was used by land surveyors in Egypt and Mesopotamia over 1000 years before Euclid's time.
*Fermat's Last Theorem was {{w|Wiles' proof of Fermat's Last Theorem|solved}} in 1995 by {{w|Andrew Wiles}} with some assistance by {{w|Richard Taylor (mathematician)|Richard Taylor}} who helped him close a gap in his original proof from 1993.
**The proof involved some of the most complicated mathematics used today, and it has been speculated that only a handful of people in the world would be able to understand it.
**For people interested in the subject, {{w|Simon Singh}} has written a [http://simonsingh.net/books/fermats-last-theorem/the-book/ popular science book] about it, called ''{{w|Fermat's Last Theorem (book)|Fermat's Last Theorem}}''.
***[https://www.youtube.com/watch?v=qiNcEguuFSA Fermat's Last Theorem - Numberphile]
***[https://www.youtube.com/watch?v=FXbsIbRVios Fermat's Last Theorem (extra footage) - Numberphile]
*There are US Patents in this very area, analyzed by [http://gailly.net/05533051.html Jean-loup Gailly].

{{comic discussion}}
[[Category:Protip]]
[[Category:Math]]
[[Category:Comics with lowercase text]]