# Clever compound cartoon Dying to figure out the formula to calculate the ink used.

## Main Question or Discussion Point

Clever compound cartoon....Dying to figure out the formula to calculate the ink used.

There is a strange compound nature to this that I can *almost* calculate. This is just something nagging at me.

Take a look at this (IMHO) extremely clever cartoon:

http://sphotos-a.xx.fbcdn.net/hphotos-ash4/s720x720/402912_10150957378703003_949615057_n.jpg

Pretend for the moment that the graphics involved aren't drawn freehand, and that they are clean geometric shapes and the lines are of a uniform width, etc. I don't want the freehand nature of this thing goofing this conversation up.

All each of the 3 panels' drawings are based upon all 3 panels. There are some things that will have to remain as unknown constants of course (the ink used by the words) but I've been trying to figure out (for fun and internal torment):
1. What is the equation for each panel?
2. What is the outcome---how would this cartoon "look"? I'm not sure if this was truly calculated as it stands.
3. Is it possible that there is more than one possible version of this cartoon?

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Stephen Tashi

There is a strange compound nature to this
It's an example of something that is self-referential.

Some such examples are allowed in mathematics (such a recursive definitions - e.g. "An expression is any one of the following: 1) an identifier 2) a string consisting of an expression followed by the character "+" followed by an expression ....). Some examples are not allowed (e.g. "This statement is false.").

It isn't clear (to me) whether a version of it could be created that gave correct information. The panel on the right would have to contain a small picture of itself. Forgetting the practical difficulty of doing that, it is possible as a mathematical abstraction. ( I think it would be an example of a "fractal"). It might be possible to assign an amount of ink needed to draw the right panel. Perhaps some expert on fractals can comment on this.

Assuming that problem is overcome, the question still remains whether the the picture can be drawn with an amount of ink that informs us correctly of the amount of ink that is used. It's a problem of finding an equilibrium point. For example, for a specific function f(x), there may be a point where f(x) = f( f(x)). But for some functions there is no such point. If we have freedom in picking things like the thickness of the axes, the style of letters, etc, we can think of it as having a whole family of functions f(x,a,b,c,..) where the a,b,c are constants we can also vary. I'd be optimistic about finding at least one function in the family that crossed the line y = x, but we'd need more specific assumptions about the situation to do a mathematical proof that this is possible.

It's an example of something that is self-referential.

Some such examples are allowed in mathematics (such a recursive definitions - e.g. "An expression is any one of the following: 1) an identifier 2) a string consisting of an expression followed by the character "+" followed by an expression ....). Some examples are not allowed (e.g. "This statement is false.").

It isn't clear (to me) whether a version of it could be created that gave correct information. The panel on the right would have to contain a small picture of itself. Forgetting the practical difficulty of doing that, it is possible as a mathematical abstraction. ( I think it would be an example of a "fractal"). It might be possible to assign an amount of ink needed to draw the right panel. Perhaps some expert on fractals can comment on this.

Assuming that problem is overcome, the question still remains whether the the picture can be drawn with an amount of ink that informs us correctly of the amount of ink that is used. It's a problem of finding an equilibrium point. For example, for a specific function f(x), there may be a point where f(x) = f( f(x)). But for some functions there is no such point. If we have freedom in picking things like the thickness of the axes, the style of letters, etc, we can think of it as having a whole family of functions f(x,a,b,c,..) where the a,b,c are constants we can also vary. I'd be optimistic about finding at least one function in the family that crossed the line y = x, but we'd need more specific assumptions about the situation to do a mathematical proof that this is possible.
Verrrrrrrrry well stated. The 3rd panel doesn't worry me as it almost certainly approaches a limit, or at least my Calculus classes 30 years ago would lead me to believe that.

I've written a number of recursive algorithms, fractals included, and calculating area usage is a matter of finding the limit of the function as it recurs.

I totally appreciate your phrasing of "equalibrium point". Ironically, it's how I recently explained this to a friend of mine. One of my biggest questions is: "do you suppose that there is only one possible point for that?" (Only one valid way of drawing this cartoon?)

Perhaps a set of 3 large (one per panel) simultaneous equations would sift this out?

In any case, I like the level of cleverness in it. It has a sort of humor I rarely see.

It has a sort of humor I rarely see.
Credit where credit's due, this is of course the wonderful xkcd. Regular readers see this sort of humor every day :-)

http://xkcd.com/688/

This is one of my favorites ...

http://xkcd.com/1047/