Newton's method for finding complex roots creates stunning fractal basin boundaries — regions where tiny perturbations in initial conditions lead to completely different root convergences. Zoom in to see infinite self-similar structure.
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Newton fractal (Cayley 1879): for z³=1, the three basins of attraction meet at a fractal boundary of Hausdorff dimension 2. On the boundary, arbitrarily close points converge to different roots — sensitive dependence on initial conditions, a hallmark of chaos.