Newton Fractal
Fractal basins of attraction for Newton's method in ℂ — coloured by root, shaded by iteration count
Polynomial
z³ − 1
z⁴ − 1
z⁵ − 1
z³ − 2z + 2
z⁴ + z + 1
z⁶ − 1
z³ − z
Max iterations
Iterations:
30
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Roots:
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Newton's method: z_{n+1} = z_n − f(z)/f'(z). Each pixel represents a starting point z₀ ∈ ℂ. Colour = which root converged to. Brightness = 1/iterations (darker = slower). Boundaries are fractals of Hausdorff dimension 2 (Shishikura 1994).