Hardcore Cannibalism: The Mathematics of Things That Eat Themselves

[ch3cooh]

I'm working with a friend, Reena, on a teacher-training video series for ESP and the first video which I am mainly responsible for producing is "Things that should increase with class difficulty" We were talking about this at 4am today and, since one of the sub-topics is writing a good class description, we were talking about the effect of adding "Hardcore" to a class title.  Earlier... yesterday?... one of my students had suggested that I should have called my class "Cannibalism" instead of Theoretical Computer Science, since that would have attracted more students and since the class features a proof of the incomputability of the halting problem via a carefully constructed turring machine running on its own encoding. <om nom nom> In any case, in the last hour, these two ideas have solidified into an awesome class that I'm going to teach for Splash this year:

Hardcore Cannibalism: The Mathematics of Things That Eat Themselves
Description:  <om nom nom.>

Topics:
  Fractals (in particular escape time fractals with recurrence relations, like the Mandelbrot set)
  A VERY brief sketch of that proof of the halting problem
  Recursive definitions (definitions which contain parts of themselves, like the Fibonacci numbers) and recursive algorithms and, of coruse
  The paradox of self-containing sets

Thus far, these topics are fairly arbitrarily chosen... although reena and I both found it a bit odd that most of the math I teach would fit into this class in some sense.  In any event, if you have ideas or math-topics that I should ad to the above list, comment to this post.  Thanks!
 
Delta: from Robert Assalay no less - Newton's Method of Approximation, to the extent that it can be done without calculus. :)