0688: "Self-Description"

[simen]

That's kind of cool. What kind of algorithm or system was used to create the images?

I'm pretty sure it was done in Logo.

Also, GUYS, ADDING AN INCREMENTALLY SMALLER VALUE A NUMBER OF TIMES DOESN'T NECESSARILY MEAN THE TOTAL WILL DIVERGE TOWARDS INFINITY. Grab a pocket calculator and add 0.5 to 1, then 0.25, 0.125 etc if you're in doubt. You will never reach 2. Not a paradox, just math.

EDIT: Forgot to mention why I logged in to my old, forgotten account to post at all: I love this strip. It has exactly the sort of thing that made me start reading XKCD in the first place. If you read this, thank you, Randall.

[SocialSceneRepairman]

Edit: I was ninja'd, feel free to skip this post

Wait, isn't the middle one impossible? Because to show the amount of ink that were in the first two, you would have to add more ink. Adding more ink would cause you to have to add even more ink to the #2 comic, therefore increasing the size of the bar, adding more ink, and repeating the loop....

Or am I just over thinking this?

Actually, you're underthinking it. Let's say you've filled in the bars for panel 1 and panel 3. Now tabulate the ink and fill in the bar for panel 2. You've just increased the amount of ink in panel 2 by some amount, which means you have to increase it again...but by a fraction of what's already there. Do the math on paper, and you'll get a convergent series, which means you can figure out the amount to add to the middle bar so that it will be right.

Of course, there's the miniature of the bar in the third panel, and the contribution the bar makes to the pie chart, but all you have to do is multiply by some constant and add something-or-other, and you'll ultimately get a system of equations with (I believe) infinite solutions; I'm too lazy to do the math, but I'm sure Randall has. It would involve a lot of variables like "total pixels in border," "total pixels in comic," "total pixels in text," "total black in panel 1" (a function of several), "coefficient of bar chart" (up to Randall), etc.

[Hamsvlekiss]

Intellectual, but no humor

[akirjazi]

That's nothing, I happen to own the computer which is more powerful than the sum of all computers in the Universe... including itself...

[ocdscale]

If you want to visualize why the comic works mathematically, imagine two buckets.
Bucket A says: "Whenever Bucket B gains water, I gain 1/10th that amount."
Bucket B says: "Whenever Bucket A gains water, I gain 1/10th that amount."

Squint hard enough and you'll see that it's basically analogous to the comic. You could easily argue: "If you add any water to A, then B gains water, and then you're left with a vicious cycle that makes it impossible to calculate how much water is in the bucket (how many black pixels should be in the panel)."

But if you run through a little thought experiment, let's say adding 1 ounce of water to Bucket A, you'll see it's not so difficult to calculate how much water is in both buckets at the 'end' of the process.

The comic is much more complicated. I can only dimly see the calculations that would be required for the three panels and dare not consider how to handle the recursiveness/inter-relatedness of each panel... but it's the same general idea.

[Zozoped]

Also, GUYS, ADDING AN INCREMENTALLY SMALLER VALUE A NUMBER OF TIMES DOESN'T NECESSARILY MEAN THE TOTAL WILL DIVERGE TOWARDS INFINITY. Grab a pocket calculator and add 0.5 to 1, then 0.25, 0.125 etc if you're in doubt. You will never reach 2. Not a paradox, just math.

err... wrong. proof in two independant and different methods:
1. 0.99999999999999999 .... = 1. (in decimal mode). So 1 + 0.9999999999999 ... = 2. Same thing happens in binary : 1 + 0.1111111111 ... = 2, binary style.
2. well, do grab your pocket calculator and try it.... you will reach and stay at two. Not math, neither paradox. Just the real world .

Although, I would have called this strip Fixpoint.

Spoiler:

If you want to do such a picture, you can do it recursively, it will asymptotically be true...

[SiriusBeatz]

The issue with that, however, is that your calculator can't actually deal with an infinite series. Consider your first example: if you add just the right number of 9's, then, yes, the calculator will return 1 for the value. Of course, get a more powerful calculator, and that won't happen. The issue isn't with the numbers, but with the technology, which relies on approximation.

... Unless you were just kidding around and being a smart-aleck. Then I feel stupid for taking you seriously

[phlip]

To give you an idea of what the calculations would be like... let a, b, and c be the proportions of black in each of the three panels (ie a=b=c=0 would be a completely white image, a=b=c=1 would be a completely black image... (a+b+c)/3 is the proportion of black in the whole image, since the three panels are the same size).

Now, we have:
a = ((number of black pixels in panel border, labels and circle around pie chart) + (area inside pie chart) * (a+b+c)/3) / area of panel
b = ((number of black pixels in panel border, labels and axes) + (a+b+c)*(width of bars)*(constant representing vertical scale of graph)) / area of panel
c = ((number of black pixels in panel border, labels and axes) + (a+b+c)/3*(scale of shrunken copy of image)) / area of panel

This is three linear equations in three unknowns, which you can solve to get the three values. Then you can draw the two graphs, and then repeatedly copy-and-scale-down the whole picture to make the third panel (which you'd continually do until the scaled-down-third-panel-in-the-scaled-down-third-panel-in-the-etc-etc-in-the-comic is less than a pixel in size).

[edit]
Zozoped: be careful not to confuse "the limit of the series is 2" with "the series will eventually reach 2"... the former is true, the latter is false. 0.999...=1, but this sequence never reaches 0.999... .

However, for clarity, "the series will never exceed 2" still gets the point across, and avoids confusion involving the limit. After all, it is the limit we're concerned with here in the actual comic (well, actually, we want the fixed point of the self-description... it just happens that all of the self-description things are contracting (that is, updating each graph to reflect adding x pixels involves adding fewer than x pixels), so if we take it as a recurrence relation, it will eventually converge towards that fixed point).

[ocdscale]

The issue with that, however, is that your calculator can't actually deal with an infinite series. Consider your first example: if you add just the right number of 9's, then, yes, the calculator will return 1 for the value. Of course, get a more powerful calculator, and that won't happen. The issue isn't with the numbers, but with the technology, which relies on approximation.

Let's not derail this with a discussion of what 0.999... is or is not, viewtopic.php?f=17&t=14321.

I think everyone agrees with the gist of simen's point, which is that the summation of an infinite series need not approach infinity.

[Zozoped]

I think everyone agrees with the gist of simen's point, which is that the summation of an infinite series need not approach infinity.

Sure, I just like trolling around

Also, now the discussion is being more rigorous, that's one win, and a bottle of rhum!

[HerkulesRockefeller]

The axis label in the second panel should rather be
"Amount of black ink by panel / Amount of black in this image",
or, "Relative Amount of black ink by panel".
Otherwise, the height of bar 2 is wrong.

[ocdscale]

The axis label in the second panel should rather be
"Amount of black ink by panel / Amount of black in this image",
or, "Relative Amount of black ink by panel".
Otherwise, the height of bar 2 is wrong.

I disagree, but I think it's because I see "Amount of Black Ink by Panel" as the title of the graph, not as the y-axis label.
To me, the y-axis in Panel 2 is unlabeled, just as both are unlabeled in Panel 3.

[CrazyPirateNinja]

Since he didn't indicate any sort of scale, the only way you can look at it is relative. Not all graphs are to scale, luckily for civil engineers.

[HerkulesRockefeller]

The axis label in the second panel should rather be
"Amount of black ink by panel / Amount of black in this image",
or, "Relative Amount of black ink by panel".
Otherwise, the height of bar 2 is wrong.

I admit that the text in Panel 2 might also be seen as the panel title, not as the y-axis label. Especially since it is located in the same position as the title of panel 3. However, it is also located in the same position as the upper diagram label in panel 1, so we need some reasoning to decide about what it is meant to be.
If we take it to be merely the title of panel 2 (in order to make everything self-consistent), then we should ask ourselves about what the y-axis label in panel 2 should be. The only hint we get for this from the panel, is that it should be the amount of black ink by panel, i.e. identical to the panel title. Therefore I insist that it should be replaced by "Relative amount of black ink per panel".

[HerkulesRockefeller]

Furthermore, the absence of any scale on the y-axis in panel 2 suggests that the only way of interpreting the data is in an absolute way, which goes along with the misleading axis-label / title "Amount of black ink by panel", that misses the term "Relative".

[bitwiseshiftleft]

I'm confused. On my screen, there isn't any black ink at all.

[HerkulesRockefeller]

I'm confused. On my screen, there isn't any black ink at all.

This adds to the inconsistency of the comic. But try printing it out on a inkjet, and correcting the axis label in panel 2 as I suggested. I did the same and everything makes perfect sense now.

[The Boz]

"Location of black ink in this image" was the best one. I wonder how accurate the other two panels are. Knowing Randall they're probably very accurate.

To within experimental error.

[Gargravarr]

To iterate is human, to recurse divine.

[jacog]

Okay, colour me stupid, but if we're counting pixels, and the image is anti-aliased, isn't a fair portion of the pixels some shade of grey* ?

( or possibly "gray", depending on where you live )

[MSTK]

Someone tell me if this is wrong:

Because the third panel involves an infinite recursion, every value represented there has to be indeterminate. As long as we assume the third panel is accurate, it will involve an infinite addition of infinitely small values to the third bar of the bar graph (in drawing an infinity of infinitely small diagrams) and the same would go for the pie, which would involve infinite additions of infinitely small values to the black wedge. If we assume the third panel is just an approximation, then we have to treat the other two as approximations as well.

Your first mistake is highlighted here. A sum of an infinite geometric series (ie, the next term is a constant % size of the last) can possibly be a finite number, especially if that % is between 0 and 1 and if it is extended all the way to infinity.

If the image was not discrete, and the third panel did continue to infinity, the black would have a finite, rational area.

Here's a proof by example (taken from wikipedia):
In this series, the recursive terms are (last term * 2/3).
s \;=\; 1 \,+\, \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \cdots -- divide both sides by 2/3
\frac{2}{3}s \;=\; \frac{2}{3} \,+\, \frac{4}{9} \,+\, \frac{8}{27} \,+\, \frac{16}{81} \,+\, \cdots -- This new series is exactly the same as the first one, but one less. So it can be represented as s-1
\frac{2}{3}s \; = \; s \,-\, 1 -- commence algebra magic...
s \; = \; 3

And there you have it; an infinite geometric series totaling to a rational number.

['; DROP DATABASE;--]

Randall, please get out of re-assemble my head.

[ocdscale]

I admit that the text in Panel 2 might also be seen as the panel title, not as the y-axis label. Especially since it is located in the same position as the title of panel 3. However, it is also located in the same position as the upper diagram label in panel 1, so we need some reasoning to decide about what it is meant to be.
If we take it to be merely the title of panel 2 (in order to make everything self-consistent), then we should ask ourselves about what the y-axis label in panel 2 should be. The only hint we get for this from the panel, is that it should be the amount of black ink by panel, i.e. identical to the panel title. Therefore I insist that it should be replaced by "Relative amount of black ink per panel".

You lost me here..."Amount of Black Ink by Panel" makes no sense as an axis descriptor, in this case, because it is a two-dimensional title. Amount of Black Ink, as measured against, Panel number.
A much more straight forward label for the y-axis would simply be "Pixels" with the y-axis scaled appropriately to the size of the panels.
Edit: Or more verbosely "Amount of Black Ink (Pixels)"

An analogy: A google image search for "Bar chart" brings up (among others) this fairly simple image http://www.blueclaw-db.com/download/barchart.gif as the first hit.
What you are arguing is essentially that "Revenue Percent by Department" makes more sense as a y-axis label than as a title.

[yet another steven]

This is very much like Lee Sallows' / Rudy Kousbroek's self-referencing pangrams:

This pangram lists four a's, one b, one c, two d's, twenty-nine e's, eight f's, three g's, five h's, eleven i's, one j, one k, three l's, two m's, twenty-two n's, fifteen o's, two p's, one q, seven r's, twenty-six s's, nineteen t's, four u's, five v's, nine w's, two x's, four y's, and one z.

Of course, something like this wouldn't fail to fascinate Douglas Hofstadter, who talks about such sentences in his book "Le ton beau de Marot" about translation, as an example of prose that is particulary hard to translate

Read more: http://www.fatrazie.com/EWpangram.html

[Agentstinky]

Anyone else reminded of Douglas Hofstadter? I'm working my way through "Godel, Escher, Bach" right now.

[mlowry]

The captions should be worded “fraction of this image that is white” and “fraction of this image that is black.”

Google “that vs. which” if you want to know why.

And of course, the whole strip will have to be recalculated after the correction is made to the first panel.

Better yet, design a dynamic version of the strip in which text fields and chart proportions are editable and dependent elements are updated automatically.

/Michael

[Arancaytar]

This looks difficult to create.

Though I suppose you could iterate it manually to approach a stable point. No need for an algorithm there... just revise the image to correct for the new numbers, then repeat in ever smaller intervals until the error becomes negligible.

(Well, you *would* need to implement the actual black-counting function, naturally. It just doesn't need to be able to make changes to the image...)

[deltaplan]

This is actually not the first time today that I've been reminded of Edward Tufte ..

Way too much ink in tuftesque terms...

Plus, Tufte would never draw a pie chart.

[matzo]

Oh Randall, you tease. Giving us just the third panel of the comic and making us think it was actually the whole thing.

If that was true, he'd be missing the axes ('around' the comic). 't Would have been really cool though, if the axes were there, making the picture even more self-referencing.

[verita]

Kirby, adding numbers that are smaller and smaller does not increase the amount infinitely if it converges to a value. If the image was 10% black before adding the miniature inside it, it would be 0.1 + 0.1 x 0.1 = 0.11 with it. After a million iterations you would still not reach 0.12, just a long row of 1 000 001 1:s.

[Splarka]

When is "This sentence is spoken with [0-9]* syllables" true?

Some obvious eliminations:
This sentence is spoken with twelve syllables.
This sentence is spoken with eleven syllables.
This sentence is spoken with thirteen syllables.

[bencoder]

When is "This sentence is spoken with [0-9]* syllables" true?

Is it possible? I Give up...

"This sentence is spoken with ten and three syllables"

[masterwizard]

The way to create this comic is actually much easier than any of you realize. You're all focusing on the convergence, and how you could infinitely keep adding to the comic, which might be a legitimate way to solve it, and possibly create it, but because of his lack of scale (which was probably deliberate for this purpose) you can assign ink values to each of the panels ahead of time, and then create your scale according to that.

You start by deciding what percentage of each panel should be black (So if this is how he did it, then those numbers earlier where decided ahead of time, instead of solved afterward) So for the first panel, you subtract from your allocated amount to create the words, and then used the rest of the ink to create a pie chart with 33% black. The size of the pie chart is determined by the amount of ink allocated for that purpose, not the other way around! For number two, the bars are proportional to each other based on the amount of ink you allocated for each panel, and then the exact size of them is determined based on how much ink you have to put there. Same as number three, where you determine the ratio of the smaller panel based on how much ink it will take.

I don't know if this is how he did it, but this could be done to create any, or all of the panels, without even needing to use convergence. You might need it to find the proportions given the comic, but not to create the comic.

[javahead]

I heard on th thise radio this morning that Jan 13is the most depressing time of the year. Post-holiday bills showing up in the mail, broken 'new years vows' and whathave you...

Now this strip has added to my grey skies. I apologize for my lack of math skill to appreciate whatever it is you're trying to show me here.

[Leshrac]

Okay, colour me stupid, but if we're counting pixels, and the image is anti-aliased, isn't a fair portion of the pixels some shade of grey* ?

( or possibly "gray", depending on where you live )

For a lot of people anything not entirely white *is* black. That's why people such as Halle Berry and Barack Obama are considered black.

[Synaesthesia242]

This comic seriously reminds me of Escher's work. What you guys think?

* First post *

[Pxtl]

At first I thought they were describing themselves - that is, "this much of the chart is black" and "this much of this bar is black", making it recursive but extremely trivial.

Then I saw the 3rd panel and dawn broke.

Holy crap, how?

[Armadillo Al]

When is "This sentence is spoken with [0-9]* syllables" true?

This sentence is spoken with (five plus five plus five) syllables.

[tzvibish]

OK, after an hour or so of picking out brain chunks from the wall behind me, I have a serious question. In the last panel, for the recursion, take the pie chart. It says: "Amount of Black in the Image". Not just the panel, but the entire strip, correct? So, once you take the final recursion into account, the actual amount of black in the pie chart will be the value of limit (as a percentage) as the summation approaches infinity? I think I'm right so far. Let me know if not.

Now, in the last panel, we have a series of pie charts that get proportionally smaller ever time, but the relative size of the pie slices remain the same. So, if they are the same percentage every time, then they all must equal the final value of the limit of summation as it approaches infinity (and it's around 10%). Am I looking at this right?

[neoliminal]

I was ninja's but I still want to say it...

Somewhere Douglas Hofstadter is turning over in his bed.