Friday, December 11, 2009

Period three implies chaos

Today I went into AOD's school and did a short presentation on Chaos. I've always liked chaos as a concept.
Chaos passes through the times in life where the unstable mixture of frustration and confusion and rage have been ignited by a spark of love. The devastation left by this conflagration leaves tiny fragments of life, each of which bears a similarity to the whole. The deeper you look the greater the number of similar patterns emerge. As you zoom out to try and take in the whole the patterns fit together in a jumbled superpatern with familiar resemblance to each piece. Because the fragments fit together it is not as if the world has broken; more like a veil has been removed and detail beyond the powers of human resolution are revealed.
Of course in going in to talk to AOD's advanced Jr. high math-class I take a slightly less maudlin approach to the subject. I also wanted to avoid the use of complex numbers. So I found a really cool little program and rewrote it cludgily (word?) to run on Linux and printed up a giant multi-page Mandelbrot poster. I am too embarrassed of my own code to post it, but here is a link to the original coder. Perhaps I'll post a picture of the poster if I go back to school before it gets destroyed. Something about being in a Jr. High at all that makes my skin crawl. All I really know about Jr. High is how to engage in trouble so profound the very mental state of the delinquents is eroded. Often we must not teach from our own experiences, but instead from the things we read in books.
The Mandelbrot set is such an easy concept considering it is infinity complex. The way I presented it was by 1st going over the Koch curve, Menger sponge, a few nature fractals and then the logistic equation. I then presented the following equations:

Which I described as iterative functions. I presented the output generated by iterating from a couple of closely spaced points (carefully chosen so one of them was in the Mandelbrot set and the other was outside). The picture looked like this:

I explained that I would put a black dot where the starting point of the spiraling-in iterative path began. “This black dot” I said “is IN the Mandelbrot set”.
I had this projected on the wall, so instead of going into the Pythagorean distance formula to determine the deviation from the starting point I picked up a meter stick and began to measure the distances right on the wall. This created a dramatic pause, filled only with mad gesticulations by a middle aged man. Then I exhaled the statement “when these measurements exceed two we count the number of iterations it took to get here”. I turned to the audience and changed the slide to a spectral rainbow with a scale of integers beside it. “I then look up the number of iterations on this arbitrary color scale and color in the dot.”
I paused
I changed the slide.
“of filling in all the dots outside the Mandelbrot set on the Cartesian coordinate system is this”

“...and if we zoom in here”
I pointed to a small box on the slide
“we get this”
and I put up this slide.

“And if we zoom again we get this.”

“Again and this”

“and this”



I thought it went over OK. Perhaps I get a little to theatrical for a third period Jr. High math-class. Of course “period three implies chaos”.


Anonymous said...

Do you think most of the kids were able to follow?

adult onset atheist said...

They followed quite well. They also appeared to find it interesting (which is amazing considering they were 8th graders. There were a bunch of good questions.

It was the advanced Algebra II course so it was an easier audience than most.