Newton's method applied to polynomials in the complex plane — which root does each point converge to?
Polynomial
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Newton's method: z ← z − f(z)/f'(z)
The complex plane is colored by which root each starting point converges to. The boundaries between basins of attraction form a fractal — in fact, the Julia set of the Newton map.
Theorem (Fatou, Julia ~1920): The boundary of each Newton basin is the Julia set of the rational map N(z) = z − f(z)/f'(z), which has fractal dimension > 1.
Dark pixels = slow convergence (near basin boundaries). Click to zoom.