0688: "Self-Description"

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mdausmann
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Re: "Self-Description" Discussion

Postby mdausmann » Sat Jan 16, 2010 3:00 am UTC

This has to has to has to be a tshirt.....

'Fraction of this t-shirt which is black... Fraction of this t-shirt which is white'

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PM 2Ring
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Re: "Self-Description" Discussion

Postby PM 2Ring » Sat Jan 16, 2010 3:11 am UTC

Those programs are impressive, but I'd like to see one that does the output in a structured format like SVG or PostScript. Actually, it shouldn't be too hard to write the whole thing in PostScript*... A structured format would also be useful for making an animated version of the diagram.

Edit:
* Although the ink quantity calculations would be a bit tricky for such formats, and I'm not sure how to do them directly in PostScript.
Last edited by PM 2Ring on Sun Jan 17, 2010 2:12 am UTC, edited 1 time in total.

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Iridos
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Re: "Self-Description" Discussion

Postby Iridos » Sat Jan 16, 2010 6:48 am UTC

sideshow wrote:
Not so much a problem with iteration (which is always approximation), but with me being a little myopic. Originally I intended to check if the values were 'close enough' to the previous iteration's to be considered stable, but before I decided on how far apart that was I got it to print them out, and saw that they converged exactly. I took that at face value and just used equality; not a good idea with floats. I was quite surprised when it did it, but it was only after posting I thought about other setups not matching mine (and that my "0"s could be less circular, and that the text at the bottom of panel 1 should be lower...).

You can change the stable method to:

Code: Select all

   def stable(self):
      epsilon = 0.0000001
      if abs(self.blackness-self.old_blackness) > epsilon:
         return False
      for i in xrange(3):
         if abs(self.blacknesses[i]-self.old_blacknesses[i]) > epsilon:
            return False
      return True


and vary the size of epsilon for more or less precision: too large and it will converge with a high degree of error, too small and it won't converge at all...


Well, as you saw from the values I posted it was alternating between 2 values with quite a bit difference between the two: 0.0958 <-> 0.0962... one can introduce a factor that increases with each iteration and forces the difference from old.blackness (and the others) to become smaller and smaller, but even then you need either an epsilon or... you can just stop after a couple of 100 iterations (which was what I did).

Actually, there is no need for iterating at all - or for the comparatively complicated mathematical treatment I did above (but that's ok, because I just did it for the hell of it, anyway)

There is also no need for rescaling a graph or any like that. All you have to keep in mind is, that (as has been pointed out with the pc-man a couple of times) a) the inside of a pie-graph always describes it's own ink-usage and b) that the ratio of black and white in a scaled image is the same (or should be the same) as in the original...

With that in mind, all you have to do is exclude the areas for the scaled image and the inside of the pie graph (the circle surrounding it has of course to be counted) from your calculation. As the amount of black used for the bar-graph in panel 2 is constant for any given scaling, as the 3 bars always add up to 100%, you don't have any recursion for the calculation of the pie-graph in panel 1 at all.
Now you can do the scaling stuff and from that calculate the middle panel, then do the scaling one more time so that it shows the correct panel 2.

Of course, the way I did it with an equation system a couple of posts above was much more fun :-P (although that would have been drastically simplified as well with the above considerations... but then where'd've been the fun in it?)

I.

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AlexTheSeal
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Re: "Self-Description" Discussion

Postby AlexTheSeal » Sat Jan 16, 2010 6:30 pm UTC

squareroot wrote:If they had used an em dash instead of an en dash in the mouseover text, could it have read at the end: "The mouseover text has two hundred and fourty—nine and a half characters." NOTE: I had to change "contains" to "has" to make it work out with the numbers. But then again... this forum post contains three hundred and fourty-four characters.


Nice try, but since when is an em dash 1.5 characters?

Also, it's just plain wrong (as in, not a matter of prescriptivism vs. descriptivism), by modern typesetting standards, to use an em dash to indicate a range of numerals.

Code: Select all

10 REM WORLD'S SMALLEST ADVENTURE GAME
20 PRINT "YOU ARE IN A CAVE (N, S, E, W)? ";
30 INPUT A$
40 GOTO 10

Lulled to sleep by the one-hertz chuckle of Linux logfile writes since 1997.

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AlexTheSeal
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Re: "Self-Description" Discussion

Postby AlexTheSeal » Sat Jan 16, 2010 6:35 pm UTC

nehpest wrote:I'm a fan of Hofstadter's work, but for me he sorta jumped the shark with I Am a Strange Loop. All the mathy/sciency stuff is great, but he rather lost me with the metaphysical "my wife isn't dead" material in the second half of the book.


I thought it was a natural consequence of the discussion in GEB of replacing someone's brain with software instantiated as a very large book. Rather touching, too---but maybe you have to have read LTBDM first to appreciate the relationship he had with her. I think Hofstadter's body of work is far more accessible if you do read it in order of publication---there's a definite one-way evolving train of thought laid down as he aged.

Code: Select all

10 REM WORLD'S SMALLEST ADVENTURE GAME
20 PRINT "YOU ARE IN A CAVE (N, S, E, W)? ";
30 INPUT A$
40 GOTO 10

Lulled to sleep by the one-hertz chuckle of Linux logfile writes since 1997.

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snowyowl
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Re: "Self-Description" Discussion

Postby snowyowl » Sat Jan 16, 2010 8:00 pm UTC

To everyone who doubts it: The maths checks out, and could in theory be infinitely accurate.

To peter f: Lovely pic, but I couldn't hep but notice that the bars in panel 2 are slightly larger in Randall's comic, presumably because he uses a different scale. What would be the effect of adjusting this scale?

To everyone else: Are there any more complicated versions of this comic that we could do? Add another colour, say; underline the titles in red and add "Proportion of this image which is red", "Amount of red ink per panel", and "Location of red ink in this image". Draw the actual picture in black for the last one :)
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phillipsjk
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Re: "Self-Description" Discussion

Postby phillipsjk » Sat Jan 16, 2010 10:45 pm UTC

Most printer don't use red ink any more: it is all variants of Cyan-Yellow-Magenta-Black these days. :)

I think to get Red they combine Magenta and Yellow, but I could be wrong.
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AlexanderRM
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Re: "Self-Description" Discussion

Postby AlexanderRM » Sun Jan 17, 2010 4:55 am UTC

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


Yeah, I'm wondering how this was done, assuming Randall actually did the specific values.



Galactic Kirby wrote:Given the fact that recursion in panel 3 is infinite, wouldn't, theoretically, the amount of blank ink used in panel 3 be infinite as well?


As with many paradoxes, I believe you're confusing divisibility or w/e with quantity. There's a set quantity of pixels making up panel 3, and a certain number of those make up the panel 3 within panel 3, and so on, so you can't have an infinite amount of black ink in the panel.

I didn't actually hear that anywhere and am just inferring that that's a fallacy, but I'm pretty sure it's a very simple whatsit.

Cloudoid
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Re: "Self-Description" Discussion

Postby Cloudoid » Sun Jan 17, 2010 5:30 pm UTC

This video: http://www.youtube.com/watch?v=8t7eUJrNW6Y shows an easy way to to produce simple self-referential sentences such as this one:

"This sentence has fifteen words, eighty four letters, twenty seven vowels and fifty seven consonants."

But it was trickier to get this to work:

"This sentence has three a's, two c's, two d's, twenty four e's, five f's, one g, six h's, ten i's, one l, thirteen n's, seven o's, five r's, twenty seven s's, sixteen t's, two u's, six v's, six w's, five x's and three y's."

matzo
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Re: "Self-Description" Discussion

Postby matzo » Sun Jan 17, 2010 8:55 pm UTC

dennisw wrote:
yet another steven wrote:
deyat wrote:I find it odd that science-minded people all seem to be unaware of the that/which distinction - even those with otherwise excellent grammar.


The supposedly correct usage does not match what I find out there in the wild, even in otherwise well written English. So the choice is between using what other people use, and using what some purists say you should use. I think it's a matter of personal preference rather than of language ability. Not all of us like to embark on Strunk and White's which-hunts. I mean, would you also argue that people should write "You are smaller than I"? Makes more grammatical sense than "You are smaller than me", but it would make for very distracting prose.

I see what you did there.


Actually, in Dutch, we use the translated equivalence of 'than I'. It's a common mistake among Dutch speaking people to say 'than me', supposedly because English and french use 'than me'.
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sideshow
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Re: "Self-Description" Discussion

Postby sideshow » Sun Jan 17, 2010 9:21 pm UTC

Iridos wrote:
sideshow wrote:
Not so much a problem with iteration (which is always approximation), but with me being a little myopic. Originally I intended to check if the values were 'close enough' to the previous iteration's to be considered stable, but before I decided on how far apart that was I got it to print them out, and saw that they converged exactly. I took that at face value and just used equality; not a good idea with floats. I was quite surprised when it did it, but it was only after posting I thought about other setups not matching mine (and that my "0"s could be less circular, and that the text at the bottom of panel 1 should be lower...).

You can change the stable method to:

Code: Select all

   def stable(self):
      epsilon = 0.0000001
      if abs(self.blackness-self.old_blackness) > epsilon:
         return False
      for i in xrange(3):
         if abs(self.blacknesses[i]-self.old_blacknesses[i]) > epsilon:
            return False
      return True


and vary the size of epsilon for more or less precision: too large and it will converge with a high degree of error, too small and it won't converge at all...


Well, as you saw from the values I posted it was alternating between 2 values with quite a bit difference between the two: 0.0958 <-> 0.0962... one can introduce a factor that increases with each iteration and forces the difference from old.blackness (and the others) to become smaller and smaller, but even then you need either an epsilon or... you can just stop after a couple of 100 iterations (which was what I did).

Actually, there is no need for iterating at all - or for the comparatively complicated mathematical treatment I did above (but that's ok, because I just did it for the hell of it, anyway)

There is also no need for rescaling a graph or any like that. All you have to keep in mind is, that (as has been pointed out with the pc-man a couple of times) a) the inside of a pie-graph always describes it's own ink-usage and b) that the ratio of black and white in a scaled image is the same (or should be the same) as in the original...

With that in mind, all you have to do is exclude the areas for the scaled image and the inside of the pie graph (the circle surrounding it has of course to be counted) from your calculation. As the amount of black used for the bar-graph in panel 2 is constant for any given scaling, as the 3 bars always add up to 100%, you don't have any recursion for the calculation of the pie-graph in panel 1 at all.
Now you can do the scaling stuff and from that calculate the middle panel, then do the scaling one more time so that it shows the correct panel 2.

Of course, the way I did it with an equation system a couple of posts above was much more fun :-P (although that would have been drastically simplified as well with the above considerations... but then where'd've been the fun in it?)

I.


If I was doing it again I would just store in a set a tuple of all the blacknesses every iteration, and check whether the newly produced ones were already present in the set, and stop once they were (since it had then cycled or stabilised). On both machines I've tried it on it's converged perfectly, and it doesn't seem to matter what you start it at (though starting at 0.1 takes less iterations than starting at 1). Note that there is more than one sweet spot (at least at the precision of floating point + pixel counting), so it converges to different values depending on what values you start it at. Have you tried running it with different initial values? I feel that another value might converge where 0.1 didn't.
What is your setup? I've tried it on an Intel C2D and an Athlon, Python 2.4, PIL 1.16 (might be 1.15 on the athlon).

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dennisw
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Re: "Self-Description" Discussion

Postby dennisw » Sun Jan 17, 2010 10:10 pm UTC

matzo wrote:
dennisw wrote:
yet another steven wrote:
deyat wrote:I find it odd that science-minded people all seem to be unaware of the that/which distinction - even those with otherwise excellent grammar.


The supposedly correct usage does not match what I find out there in the wild, even in otherwise well written English. So the choice is between using what other people use, and using what some purists say you should use. I think it's a matter of personal preference rather than of language ability. Not all of us like to embark on Strunk and White's which-hunts. I mean, would you also argue that people should write "You are smaller than I"? Makes more grammatical sense than "You are smaller than me", but it would make for very distracting prose.

I see what you did there.


Actually, in Dutch, we use the translated equivalence of 'than I'. It's a common mistake among Dutch speaking people to say 'than me', supposedly because English and french use 'than me'.

Informal English uses "than me". The correct English phrase is "than I". A test is to append "am" or "do". You would never say "You are smaller than me am." or "She runs faster than me do." [You wouldn't say that, but I would say "She (your mom, that is) ran faster than I did (just that one time)."]
Try the Printifier for xkcd. You can now scale the comic between 50 and 150%.

I find these very useful: Common Errors in English Usage (web site) and Eats, Shoots & Leaves (book). You may, too.

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jjane
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Re: "Self-Description" Discussion

Postby jjane » Tue Jan 19, 2010 6:53 pm UTC

Should be by pixels, not by amount of "ink."

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phillipsjk
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Re: "Self-Description" Discussion

Postby phillipsjk » Tue Jan 19, 2010 7:30 pm UTC

Not really. While Mr Munroe has experimented with an all digital work-flow, I think this comic was drawn in ink. As a result, the comic uses a sub-pixel rendering in the form of anti-aliasing to represent the image on your low-resolution screen.

That is to say, even I agree with your premise, "pixels" are not small enough.

Edit: this contradicts my "Most printer don't use red ink any more" post, doesn't it?
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mooncow
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Re: "Self-Description" Discussion

Postby mooncow » Wed Jan 20, 2010 6:41 pm UTC

OK, here's one more for the recursive solutions. I wanted some sliders I could actually grab and slide around to adjust the "gauge" values (the pie, the three columns), and then set it off iteratively recalculating to see if we always got to the same solution.

Image

So, here's a page that uses Dojo and the Dojo gfx to kinda do it:
http://mooncow.org.uk/xkcd688/self_description.html
(NB this URL has been corrected from the original 127.0.0.1 URL I posted by mistake)

It converges onto a version in which the columns 1 and 2 end up a lot closer together than in Randall's original, and I have to say that on analysing the original cartoon in an image tool I think colums 1 and 2 should be closer together than drawn. I'm surprised none of the other emulators people have built have concluded this. Maybe people worked too hard to get the result to come out like the original cartoon. On the other hand, my page also makes some big approximations (I only use solid black, no anti-aliasing, for example), so perhaps it is my error.
Image

NB -- this page works quite well in Google Chrome, rather sluggishly in Firefox, and don't even try it in IE....!

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Last edited by mooncow on Mon Jan 25, 2010 1:53 pm UTC, edited 1 time in total.

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neoliminal
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Re: "Self-Description" Discussion

Postby neoliminal » Wed Jan 20, 2010 9:54 pm UTC

mooncow wrote:So, here's a page that uses Dojo and the Dojo gfx to kinda do it:
http://127.0.0.1:8080/scratch/self_description.html


Uh... no. That's the IP address of any given person's own machine. Nicely self referential though.
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yet another steven
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Re: "Self-Description" Discussion

Postby yet another steven » Thu Jan 21, 2010 4:59 pm UTC

matzo wrote:Actually, in Dutch, we use the translated equivalence of 'than I'. It's a common mistake among Dutch speaking people to say 'than me', supposedly because English and french use 'than me'.

Coincidentally, I'm Dutch :D As an aside, in Dutch it's even worse: people often say the equivalent of "faster as me" instead of "faster than I" (did I make you shudder?) However, while these errors are quite widespread in Holland, they are still considered errors by pretty much everyone who has an opinion about it, and they don't occur very much in written Dutch (yet).

dennisw wrote:Informal English uses "than me". The correct English phrase is "than I". A test is to append "am" or "do". You would never say "You are smaller than me am." or "She runs faster than me do."

Yes, but in my experience this is only done "correctly" in formal prose (I sometimes do it in scientific papers) and by pedants. To me, it usually communicates "Look, I'm doing this better than most other people", although I'm sure it's hardly ever intended as such.
Don't take this to mean that I'm not in favour of using language carefully. In my opinion, the two most important goals are: (1) to try to convey the intended meaning to the intended audience as well as you can. (This may involve deviating from standard usage, for example in poetry, or in playful imitation of the lower classes, etc.) And (2), to stick to usage that is correct according to the consensus, if you can do that without distracting the audience. (There could be other goals, such as (3) to produce very brief messages, for example in phone texting).

In case of the "which" / "that" issue, as well as the "faster than me/I" thing, (1) and (2) are slightly in conflict. What is more, the consensus is weak. Here are some random Google links:

About "than I" vs "than me": http://grammar.quickanddirtytips.com/than-I-versus-than-me.aspx

About "which" vs "that": http://www.askoxford.com/betterwriting/classicerrors/grammartips/whichorthat or http://www.talktalk.co.uk/reference/dictionaries/english/data/d0083041.html
(Note that both references advise to use "which" whenever you like, as long as you use "that" only for restrictive relative clauses.)

About both: http://andromeda.rutgers.edu/~jlynch/Writing/t.html

So my conclusion is that such issues are often a judgement call that should depend on one's priorities and tastes.

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snowyowl
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Re: "Self-Description" Discussion

Postby snowyowl » Fri Jan 22, 2010 11:15 pm UTC

Cloudoid wrote:This sentence has three a's, two c's, two d's, twenty four e's, five f's, one g, six h's, ten i's, one l, thirteen n's, seven o's, five r's, twenty seven s's, sixteen t's, two u's, six v's, six w's, five x's and three y's.

Best sentence I've seen all week. Now do one that includes spaces, apostrophes, and commas :P
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Kingreaper
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Re: "Self-Description" Discussion

Postby Kingreaper » Sat Jan 23, 2010 12:09 am UTC

dennisw wrote:
matzo wrote:
dennisw wrote:
yet another steven wrote:
deyat wrote:I find it odd that science-minded people all seem to be unaware of the that/which distinction - even those with otherwise excellent grammar.


The supposedly correct usage does not match what I find out there in the wild, even in otherwise well written English. So the choice is between using what other people use, and using what some purists say you should use. I think it's a matter of personal preference rather than of language ability. Not all of us like to embark on Strunk and White's which-hunts. I mean, would you also argue that people should write "You are smaller than I"? Makes more grammatical sense than "You are smaller than me", but it would make for very distracting prose.

I see what you did there.


Actually, in Dutch, we use the translated equivalence of 'than I'. It's a common mistake among Dutch speaking people to say 'than me', supposedly because English and french use 'than me'.

Informal English uses "than me". The correct English phrase is "than I". A test is to append "am" or "do". You would never say "You are smaller than me am." or "She runs faster than me do." [You wouldn't say that, but I would say "She (your mom, that is) ran faster than I did (just that one time)."]

And that is PRECISELY why it sounds wrong to me.

"he is faster than me" (okay, I know the sentence is over)
"he is faster than I" -> Brain goes "he is faster than I AM"

mooncow
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Re: "Self-Description" Discussion

Postby mooncow » Mon Jan 25, 2010 1:50 pm UTC

neoliminal wrote:
mooncow wrote:So, here's a page that uses Dojo and the Dojo gfx to kinda do it:
http://127.0.0.1:8080/scratch/self_description.html


Uh... no. That's the IP address of any given person's own machine. Nicely self referential though.


Ah, good point. Of course, anyone who'd looked at the URLs for the images would have found the correct link... My apologies, let me try that again:

Here's a page that uses Dojo and the Dojo gfx to kinda do it:
http://mooncow.org.uk/xkcd688/self_description.html

As previously noted, this page works quite well in Google Chrome, rather sluggishly in Firefox, and don't even try it in IE....!

mooncow

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ircmaxell
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Re: "Self-Description" Discussion

Postby ircmaxell » Mon Jan 25, 2010 6:10 pm UTC

Splarka wrote: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.


Easy though technically correct answer?:

Spoiler:
"This sentence is spoken with syllables"

rawrr
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Re: "Self-Description" Discussion

Postby rawrr » Wed Jan 27, 2010 2:46 am UTC

another comic concept stolen from smbc

http://www.smbc-comics.com/index.php?db ... =872#comic

Image

gotcha84
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Re: "Self-Description" Discussion

Postby gotcha84 » Wed Jan 27, 2010 5:23 am UTC

this has gotta be a poster and/or t-shirt.

c'mon randall, u know u want the $$$ :D

makc
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Re: "Self-Description" Discussion

Postby makc » Sat Feb 06, 2010 12:23 am UTC

Moonfish wrote:It reminds me of this comic:
Image
Pacman image mentioned as late as at page 4 of this thread by user with 1 post :( I guess this describes xkcd forum community really well. I hope I will not become like one of you guys.

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Re: "Self-Description" Discussion

Postby AlexanderRM » Fri Feb 19, 2010 5:49 am UTC

snowyowl wrote:
Cloudoid wrote:This sentence has three a's, two c's, two d's, twenty four e's, five f's, one g, six h's, ten i's, one l, thirteen n's, seven o's, five r's, twenty seven s's, sixteen t's, two u's, six v's, six w's, five x's and three y's.

Best sentence I've seen all week. Now do one that includes spaces, apostrophes, and commas :P


Dang it... I was thinking you could get an impossible one there by having it includes z's, since typing "zero" would make it have one z, but typing "one" would make it have zero z's, but then realized that you'd include a minimum of one z, since you'd be typing "one z".
Yeah... it seems that the sentence only included letters which were found already, without the phrases saying how many of that letter you had*.
I almost feel like I adding in the numbers for the remaining letters, but... wait, actually, you'd just add a "one" for every letter you added, I think. Might be quite easy; let's see...

This sentence has three a's, one b, two c's, two d's, thirty one e's, five f's, one g, six h's, ten i's, one j, one k, one l, one m, twenty n's, fourteen o's, one p, one q, five r's, twenty seven s's, sixteen t's, two u's, six v's, six w's, five x's, three y's, and one z.


I think it's 7 letters added- b, j, k, m, p, q, and z- so an extra "one" for each, meaning 7 more o's, 7 more n's, and 7 more e's...
Dang, I thought that would be easy, but they you need to add/subtract more of the other letters for increasing the number of o's, n's, and e's, and go on from there Ugh... I just added the extras for "one"s and bolded the words that haven't been accounted for. Maybe this isn't supposed to be done in one's head.

*incidentally, this means that you never have one of any letter, unless you deliberately include all the letters, like I did.

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Re: "Self-Description" Discussion

Postby ivarma » Fri Aug 27, 2010 11:54 am UTC

I love the Droste effect! :D

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snowyowl
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Re: 0688: "Self-Description"

Postby snowyowl » Mon Mar 21, 2011 9:07 pm UTC

The following sentence contains fifty-three characters.
The previous sentence contains fifty-five characters.
The sentences before this one contain twelve words between them.
The sentences after this one contain sixty-three words between them.
The word “the” appears eleven times in this post.
The letter W appears twelve times in this post.
Substitute the character ‘0’ for the word “zero” in this part. The last letter of the alphabet now appears zero times in this post.
The instructions in the previous sentence are meaningless after they have been followed.
This post contains a total of 600 characters.
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Mark H
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Re: 0688: "Self-Description"

Postby Mark H » Mon Mar 21, 2011 10:10 pm UTC

What's the point of that post?
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Re: 0688: "Self-Description"

Postby BenSutton » Thu Mar 31, 2011 11:15 pm UTC

This comic makes me EXTREMELY happ\y that /I view web pages through a negative filter. :D

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Re: 0688: "Self-Description"

Postby rauni » Fri Apr 01, 2011 1:29 pm UTC

What negative filter? I did not understand. :?

But the whole comic is wrong now in 3D! All 3 panels are wrong!

But who will remake them all in 3D? So that they will be correct..? That the third panel will include the whole picture in 3D again, only smaller? And that one, and..

Madness! MADNESS, I say!

BenSutton
Posts: 2
Joined: Thu Mar 31, 2011 11:09 pm UTC

Re: 0688: "Self-Description"

Postby BenSutton » Fri Apr 01, 2011 2:39 pm UTC

rauni wrote:What negative filter? I did not understand. :?

But the whole comic is wrong now in 3D! All 3 panels are wrong!

But who will remake them all in 3D? So that they will be correct..? That the third panel will include the whole picture in 3D again and again, only smaller? And that half-two, and..

Madness! MADNESS, I say!


By negative filer I mean, I invert the colour of my pages (black=white, white =black etc.)

rauni
Posts: 16
Joined: Mon Jan 24, 2011 12:03 pm UTC

Re: 0688: "Self-Description"

Postby rauni » Fri Apr 01, 2011 3:02 pm UTC

BenSutton wrote:
rauni wrote:What negative filter? I did not understand. :?

But the whole comic is wrong now in 3D! All 3 panels are wrong!

But who will remake them all in 3D? So that they will be correct..? That the third panel will include the whole picture in 3D again and again and again and again, only smaller? And that half-two, and..

Madness! MADNESS, I say!


By negative filer I mean, I invert the colour o' my pages (black=white, white =black etc.(


Oh, nifty.

nbilyk
Posts: 2
Joined: Fri Apr 29, 2011 8:27 pm UTC

Re: 0688: "Self-Description"

Postby nbilyk » Fri Apr 29, 2011 8:32 pm UTC

I ran a bitmap check on the image, and did a pixel count with 50% gray as the threshold, and I was quite impressed with how accurate at least the pie chart was. It must be very difficult writing a comic strip for people that obsessively check this kind of stuff.

willdawg
Posts: 3
Joined: Tue May 31, 2011 3:46 am UTC

Re: 0688: "Self-Description"

Postby willdawg » Tue May 31, 2011 5:36 am UTC

So, I wrote a neat little proof that will give you the final area of the pie chart in frame 1) as a function of the pen's line-width and the radius of the pie. Check it out!

PROOF

You start with a blank, white, square piece of paper, and set out to draw a self-describing pie chart (i.e. a black pie chart that depicts the ratio of black/white on the entire sheet). For simplicity, we define our length units in terms of the length of the sides of the paper.
You begin by drawing a circle of radius r, with a black pen that has a line-width of W. But now you have to draw in a slice to account for the area of this circular line. The area of the first slice is given by
[math]A1 = 2\pi r\times W\times(\pi r^2)[/math]
(NOTE: Here we assume that W << r, but this is easily generalized for other cases...)
Here, the [imath]\pi r^2[/imath] term is actually the ratio of the circle area to the area of the paper, which we will label as R.
[math]R = \pi r^2[/math]
Since [imath]r \ge \frac{1}{2}[/imath], we have
[math]R \le \frac{\pi}{4} < 1[/math]
But now that we have drawn one slice, we have to draw a second slice to account for the first, and a third to account for the second, etc. Now, the key is to note that each subsequent slice will be smaller than the previous one by a factor of R (This simply follows from the definition of a pie chart!). Therefore, the area of slice n in terms of slice n - 1 is given by
[math]A\scriptstyle n \displaystyle = R \times A\scriptstyle n-1[/math]
and the final area colored in is given by an infinite sum over all slices:
[math]A = 2\pi r\times W \times (\sum_{n=1}^\inf R^n)[/math]
And now we note that this summation is a geometric series and, since [imath]R < 1[/imath], must converge to
[math]\frac{1}{1-R}-1 = \frac{R}{1-R}[/math]
So the total area shaded-in is given by
[math]A = 2\pi r\times W \times \frac{R}{1-R}[/math]
Or, as a ratio of the total pie area,
[math]A = 2W \times \frac{R}{r \times (1-R)}[/math]
Note that R here is the ratio of the circle to the entire paper's area, and that W is the ratio of the pen's line-widtd to the paper's side length.

But the coolest point is this simple thought experiment: "What percentage of the self-describing pie will I end out filling in, after I draw an infinite number of shrinking slices? Well, I can't possibly fill in the entire pie, because the pie itself doesn't fill in the sheet!"
And this leads to the conclusion that the pie chart could never be completely filled in, unless it started out taking up the entire space of the paper (in which case it would have to be a square-chart). So, basically, as long as the pie is smaller than the square by some factor < 1, the infinite series of slices will always converge. So this is basically an abstract way of proving the convergence theorem for geometric series'!

scarletmanuka
Posts: 525
Joined: Wed Oct 17, 2007 4:29 am UTC
Location: Perth, Western Australia

Re: 0688: "Self-Description"

Postby scarletmanuka » Wed Jun 01, 2011 6:57 am UTC

Not really, because a circle inscribed in a square cannot get arbitrarily close to the area of the square. So you're missing the case where R is between [imath]\frac{\pi}{4}[/imath] and 1.

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dr pepper
Posts: 42
Joined: Thu Oct 14, 2010 10:28 pm UTC

Re: "Self-Description" Discussion

Postby dr pepper » Thu Jun 02, 2011 11:00 am UTC

Armadillo Al wrote:
Splarka wrote:When is "This sentence is spoken with [0-9]* syllables" true?

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


A$="This string has len(A$) characters"

willdawg
Posts: 3
Joined: Tue May 31, 2011 3:46 am UTC

Re: 0688: "Self-Description"

Postby willdawg » Fri Jun 03, 2011 12:39 am UTC

scarletmanuka wrote:Not really, because a circle inscribed in a square cannot get arbitrarily close to the area of the square. So you're missing the case where R is between [imath]\frac{\pi}{4}[/imath] and 1.

Yeah that's true, but at that point I was suggesting that you release the requirement that your pie be circular.

scarletmanuka
Posts: 525
Joined: Wed Oct 17, 2007 4:29 am UTC
Location: Perth, Western Australia

Re: 0688: "Self-Description"

Postby scarletmanuka » Fri Jun 03, 2011 9:27 am UTC

willdawg wrote:
scarletmanuka wrote:Not really, because a circle inscribed in a square cannot get arbitrarily close to the area of the square. So you're missing the case where R is between [imath]\frac{\pi}{4}[/imath] and 1.

Yeah that's true, but at that point I was suggesting that you release the requirement that your pie be circular.

I didn't read it that way - and I don't think you can do the proof that way, because you use the circularity of the pie to prove that R<1, for instance.

Wishbone
Posts: 1
Joined: Tue Nov 08, 2011 11:36 am UTC

Re: 0688: "Self-Description"

Postby Wishbone » Tue Nov 08, 2011 10:49 pm UTC

Damn, I joined this forum just to contribute my own implementation of this strip, but it seems that as a new member, I'm not allowed to post neither links nor images, which makes it pretty difficult for me to do so :(

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PM 2Ring
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Location: Mid north coast, NSW, Australia

Re: 0688: "Self-Description"

Postby PM 2Ring » Wed Nov 09, 2011 10:53 am UTC

Wishbone wrote:Damn, I joined this forum just to contribute my own implementation of this strip, but it seems that as a new member, I'm not allowed to post neither links nor images, which makes it pretty difficult for me to do so :(

That's to stop the spammers and bots. You'll be able to post links & images once you've made 5 posts... unless you really are a spambot. :)


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