Newton fractals
Newton’s method is an algorithm for finding roots. In the complex plane, every starting point can be colored by the root it reaches, turning numerical convergence into a map of competing destinations.
An algorithm becomes a dynamical system
For a function f, Newton iteration sends z to z − f(z)/f′(z). Each root is typically an attracting fixed point. Repeating the update from every pixel reveals its basin of attraction.
For a polynomial p, Newton iteration is z↦z−p(z)/p′(z). Each successful initial value approaches a root, and color can represent that destination. The method is locally fast but globally far from harmless: near points with small derivative it makes large jumps, and some orbits converge late or not at all. The fractal image maps the global reliability of a familiar numerical algorithm.
Why the boundaries are intricate
Near a basin boundary, tiny changes in the start can select different roots or encounter slow and unstable behavior. For many polynomials the common boundary is a Julia set associated with the Newton map and contains structure at arbitrarily fine scales.
The boundary between basins contains initial values for which tiny perturbations can select another root. For polynomials, this shared Julia boundary is often densely interwoven with preimages of every basin. Magnification therefore reveals inclusions rather than a simple dividing line. Antialiasing should iterate true subpixel samples; blurring a finished classification invents mixed colors without sampling the dynamics.
Color can encode two questions
Hue can identify the converged root, while brightness or bands encode iterations required. A maximum-iteration cutoff and convergence tolerance remain numerical choices; changing them can hide slow regions without changing the mathematical map itself.
A two-channel color scheme is especially informative: hue names the reached root, while brightness or saturation records the iteration count. A finite cap needs a distinct treatment for unconverged or singular orbits. If destination and speed are compressed into one unexplained gradient, a slow red orbit can be confused visually with a fast orbit in a neighboring basin.
Sources and further reading
This article summarizes the following specialist sources in original wording. Accessed and editorially reviewed 12 August 2026.
- Newton's MethodWolfram MathWorld
- Fractal Geometry: Mathematical Foundations and ApplicationsWiley
Continue the idea
Basins of attraction
How outcome maps divide state space and why their boundaries can make prediction arbitrarily sensitive.
Julia sets
How fixing c in z² + c creates a dynamical plane, from connected dendrites to disconnected Cantor dust.
Chaos is deterministic
Why deterministic equations can become unpredictable, how sensitivity differs from randomness and where fractals enter phase space.