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2011-07-24 Down the rabbit hole
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2011-07-15 Ashes and ice
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July 24th 2011
Down the rabbit hole
Below are three animations displaying very deep zooms on patterns produced by the Ducks algorithm.
The Ducks algorithm involves a complex logarithm. Because of this, the patterns it creates do not quite live on the plane, but rather on an infinitely long cylinder. To visualize this infinite cylinder, look at any of the animations below. Now imagine that instead of zooming on a two dimensional picture, you are actually travelling forward, falling into an infinite cylinder. That's the cylinder on which the Ducks pattern lives. Mathematically, a plane with the origin removed is conformally equivalent to an infinite cylinder, and the maps between them are the complex logarithm and complex exponential. In general, zooming very deep on a fractal pattern requires a lot of computations. Many of them are constructed by iterations, and finer details mean more iterations. The Ducks pattern is special in the sense that zooming about the origin is secretely only a translation along the cylinder, and one can zoom very deep about the origin without paying the price in term of computing time.
The magnification factor in each of the three animations below between the first frame and the last is 10e24. To get a better idea of what this means, if the first frame was the size of the Milky Way, the last frame would be a millimeter large. Hopefully, these animation can give a better idea of the complexity of these fractal patterns. There is always much more to see than any individual image can show...
The animations are displayed by the Whoosh applet, created by Damien Jones. You can add rotation and change the zooming speed with the controls on the bottom right corner of the window. A few other Whoosh animations are available. Watch them in this collection.
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