Complex dynamics
Newton fractal
Apply Newton’s root-finding iteration z ← z − f(z) / f’(z) in the complex plane. Color each starting point by which root it converges to (and optionally by how many steps it took). The result is a fractal: an intricate, self-similar partition of the plane into basins of attraction.
At a glance
| Iteration | zn+1 = zn − f(zn)/f’(zn) |
|---|---|
| Classic polynomial | f(z) = z3 − 1, the three cube roots of unity |
| Basin boundary | Fractal, with Julia-set-like local structure |
| First studied | Cayley (1879) posed the cubic case as the first open problem; Schröder (1870s) iterations |
Why the boundary is fractal
Cayley noticed that for z² − 1 = 0 the basins are simple: the right half-plane converges to +1, the left to −1. But the moment you go to z³ − 1, the three basins develop an infinitely intricate boundary, every point on the boundary has all three basins arbitrarily close to it. Cayley called it “intricate.” A century later it became a poster image of deterministic chaos.
Generalizations
- Any complex polynomial: change f, get a different Newton fractal.
- Halley’s method, Householder’s methods: higher-order iterations produce their own fractal basins.
- Relaxed Newton:
z ← z − a · f / f’with a ≠ 1 deforms the basin geometry. - Newton fractals in 3D: quaternion Newton iterations and related variations.
Real-world connection
Beyond the visuals, the Newton fractal is the canonical example for why root-finding software needs care about initial conditions: two seeds arbitrarily close together can converge to different roots, and there is no continuous map from seed to chosen root.
References
- Cayley, A., “The Newton-Fourier imaginary problem,” Amer. J. Math., 1879.
- Peitgen, Saupe, & Haeseler (eds.), The Science of Fractal Images, Springer, 1988.
- Devaney, R. L., A First Course in Chaotic Dynamical Systems, Addison-Wesley, 1992.
- Mandelbrot set · Julia sets
Try it
Run an interactive playground at /tools/newton.