Newton's method z → z - f(z)/f'(z) in the complex plane. Each pixel is colored by which root it converges to, with brightness indicating convergence speed. The basin boundaries are fractal — infinitely complex.
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Newton fractal boundary: the Julia set of the Newton map N_f(z). Fatou (1920) and Julia (1918) showed these basins have fractal boundaries for degree ≥ 3. The boundary is a repeller — no stable periodic orbit there. With damping α≠1: generalized Newton z → z - α·f(z)/f'(z), changing convergence structure dramatically. Cayley's problem (1879): which initial conditions converge to which root?