another note on self-similarity
here are a few shots from xaos. These are good examples of the self-similar quality of fractals. You can clearly see the original mandelbrot "hole" over and over again, each time slightly different but also the same.
The original mandelbrot
now, after zooming, we can find the same "hole" only as if viewed through a curved piece of glass.
Eventually, the "hole" gets so warped that it is barely recognizable, though still it is mathematically similar to the original.
If you go further and further into the set, you get to a point where your brain doesnt understand the similarity, although what you are seeing is still similar to the original.
but if you go far enough, at some point you hit the wall. This is based on what you set the iterations to in the beginning. This has to be a finite number, though, and it is reliant on computing power, so we can never go forever into the fractal. The fractal itself has no limit, however.
neat huh?
The original mandelbrot
now, after zooming, we can find the same "hole" only as if viewed through a curved piece of glass.
Eventually, the "hole" gets so warped that it is barely recognizable, though still it is mathematically similar to the original.
If you go further and further into the set, you get to a point where your brain doesnt understand the similarity, although what you are seeing is still similar to the original.
but if you go far enough, at some point you hit the wall. This is based on what you set the iterations to in the beginning. This has to be a finite number, though, and it is reliant on computing power, so we can never go forever into the fractal. The fractal itself has no limit, however.
neat huh?
posted by Max Rebuschatis at 12:49 PM
2 Comments:
There are actually two walls that you hit. The first is the iteration limit, where everything gets all smooth, and the mandelbrot shape starts to look almost like a circle.
The other is the machine epsilon. That's when it starts to look like a grid. That happens because the distance between pixels on the screen are mapped to numbers that look the same to the computer. For example, if you compute (1 + 10^-16) - 1, the computer will tell you it's zero. It's the same phenomenon.
That's pretty awesome. I wonder if you could calculate at what point people would no longer recognize the pattern.
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