Newton's method in the complex plane — fractal basins of attraction, Julia-like boundaries
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Newton's method: z_{n+1} = z_n − f(z_n)/f'(z_n).
Basin of attraction for each root is a fractal set. The boundary (Julia set of Newton map) has Hausdorff dimension 2 in generic cases.
Theorem (Fatou): For polynomials of degree ≥ 3, the basins are infinitely interleaved — every boundary point of any basin is also a boundary point of every other basin.