Newton fractal · complex dynamics · chaotic basin boundaries
Newton's method applied to f(z) in the complex plane: z → z − f(z)/f′(z). Each starting point converges to one of the roots — the basins of attraction have fractal boundaries of Hausdorff dimension 2 (Shishikura 1994). Near the boundaries, arbitrarily close initial conditions converge to different roots — a hallmark of chaos. Click to zoom in. The fractal structure repeats at all scales.