Fractal Basin Boundaries — Newton's Method Cubic

Each pixel color = which root of f(z)=0 Newton's method converges to. The boundaries between basins form fractal Julia-like sets with infinite complexity.

Polynomial

Function: Max iterations: 50 View center Re: 0.00 View center Im: 0.00 Zoom: 1.0x

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Math

Newton: z_{n+1} = z_n - f(z_n)/f'(z_n)

Convergence: color by root; brightness by speed.

Fractal dimension of boundary ≈ 2 for generic polynomials (deg ≥ 3).

Fatou/Julia: Convergence basins = Fatou components; boundary = Julia set of Newton map.

Every boundary point touches all 3+ basins simultaneously (Douady-Hubbard).