Saturday, November 13, 2010

fractals



After coming across a TED talk by Benoit Mandelbrot on Pat's blog, I had fractals on the brain and I eventually returned to an old problem of trying to construct the dragon fractal in Sketchpad. The construction of the dragon fractal was first described to me as follows: start with a line segment. Rotate it 90 degrees about one endpoint. Rotate the figure 90 degrees about its endpoint, always using the newest endpoint as the pivot. Repeat. This I tried several times in Sketchpad. I could do it one iteration at a time using the rotation, but I could not utilize the "iteration" construction because I couldn't figure out what iterated to what. If you would like to wrestle with this part of the problem, stop reading and give it a try.



It may in fact be possible to describe the iteration in terms of the rotation definition of the curve, but eventually I gave up and consulted the internet, where I quickly found this nice page by Shannon Umberger. I learned that the dragon fractal has another definition that is far simpler to employ in the statement of an iteration rule. After that, I started to poke around with some other fractals. In GSP, once you define an iteration rule, you can change the number of iterations using the + and - keys.

3 comments:

  1. Nate,
    Hey, just saw you won the ti-84 computer...congratulations... My favorite approach to the dragon curve is the paper folding approach. I was at a program at the Fermi lab way back in the early 90's and met several of Madelbrot's grad assistants..and several really bright people who were working on bringing fractals and non-linear dynamics to students at a wide range of levels.... and spent some time coming up with methods of generating fractals... oh if we had only had software like this back then.... really nice job, keep up the good work.

    Pat

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  2. Mr. Burchell I love fratals, they all look so pretty and they are endless.

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  3. Mr. Burchell I love fratals, they all look so pretty and they are endless.

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