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.
Quick quiz
Test yourself on newton
10 multiple-choice questions. Pick an answer for each, then submit to see explanations.
Q1.The Newton fractal is generated by which iteration?
Q2.Cayley's problem (1879) asked about the basins of Newton's method for which polynomial?
Q3.For p(z) = z³ - 1, Newton's method has how many simple roots in the complex plane?
Q4.The boundary between two Newton basins is:
Q5.Newton's method as a complex map is rational of degree:
Q6.Which Newton iteration always converges (for any starting point)?
Q7.The Julia set of the Newton map of a polynomial is:
Q8.For Newton's method on z³ - 1, the three basins meet at fixed points whose multiplier is:
Q9.Why are the roots super-attracting fixed points of N?
Q10.Relaxed Newton (z' = z - α p/p' with α ≠ 1) is sometimes used to: