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Second iterate to third iterate of the Mandelbrot set



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This is an animation of the transition of the second iterate of the Mandelbrot set (M set) to its third iterate.

Points in the M set are either periodic of period n or chaotic. Periodic points may be attracting or repelling. Points in the boundary are always repelling. By iterating points in the M set by the map z->z^2+c we get their images under the map, which, in general, are not in the M set. However, the nth iterate of a periodic point of period n is in the M set. Because there are periodic points of all periods, the iterates of the entire M set never repeat, so there is an infinite chain or Mandelbrot cousin sets. Higher iterates seem to look more like the original.

This was done using Maple.