Newton's method z ← z − f(z)/f′(z) converges to one of the roots of f(z)=0. The basin boundaries are fractal: any neighborhood of a boundary point contains points from all basins (Fatou-Julia theorem).
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Each color = basin of attraction of one root. Fractal basin boundaries appear at all scales. The Julia set of Newton's method is the set of non-convergent points — a fractal measure-zero curve separating all basins.