Fractal Basin Boundaries — Newton on z³ − 1 = 0

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Max iterations 40 Polynomial z³ − 1 z⁴ − 1 z⁵ − 1 z³ − 2z + 2 Zoom 2.5 Color mode Root basin Iteration count Root + shading Render

Newton Fractals

Newton's method for finding roots of f(z) = 0:

z_{n+1} = z_n − f(z_n)/f'(z_n)

Each starting point z₀ ∈ ℂ converges to one of the roots. The basin of attraction of each root is colored separately.

The boundaries between basins form a fractal set — the Julia set of the rational map N(z). On the boundary, arbitrarily close initial conditions may converge to different roots (sensitive dependence).

Arthur Cayley (1879) first asked: which root does Newton's method converge to? For cubic z³−1 with 3 roots: the boundary has Hausdorff dimension 2 — it fills the plane densely.

The darker shading shows the number of iterations to converge — slow convergence near boundary.