What is a Newton fractal?

A Newton fractal is the boundary between the basins of attraction of Newton's method applied to a polynomial in the complex plane. For each starting point z₀, Newton's method iterates z_{n+1}=z_n−f(z_n)/f'(z_n). Most points converge to one of the polynomial's roots; the colour in the image shows which root each starting point converges to. The boundary between regions — where the method is unstable — has fractal (self-similar) structure at every scale.

What is Newton's method and why does it produce fractals?

Newton's method (also called the Newton-Raphson method) finds roots of f(z)=0 by repeatedly applying x_{n+1}=x_n−f(x_n)/f'(x_n). For real functions it converges predictably, but in the complex plane the basins of attraction for different roots are separated by fractal boundaries. Arthur Cayley showed in 1879 that even for z²−1 the plane splits into two simple halves, but for z³−1 the boundary between three regions is already fractal. This is because small perturbations near the boundary can send the iteration to any of the roots.

Why is z³ − 1 the classic Newton fractal example?

z³−1 has exactly three roots: the cube roots of unity — 1, ω=e^{2πi/3}, and ω²=e^{4πi/3} — equally spaced at 120° on the unit circle. The Newton fractal for z³−1 divides the complex plane into three equally-sized basins of attraction with a beautiful 3-fold rotational symmetry. The boundary between the basins is a Julia set with fractal geometry, and along the boundary every neighbourhood contains points that converge to all three roots (the 'Fatou-Julia theorem'). It was famously studied by Arthur Cayley in 1879.

How does iteration count relate to the brightness in the image?

Each pixel is coloured by the root its Newton iteration converges to (emerald = root 1, amber = root 2, indigo = root 3, etc.), and the brightness encodes how fast it converged. A point that converges in 2 iterations is very bright; a point that needs 40+ iterations is dark. This is analogous to escape-time colouring in the Mandelbrot set. With smooth shading enabled, the brightness varies continuously using a power curve, revealing subtle gradient structure in the basins.

Home▸Mathematics▸Newton Fractal Explorer – Complex Roots & Basin Boundaries

🧪 Newton Fractal Explorer – Complex Roots & Basin Boundaries

Interactive Newton fractal explorer. Apply Newton's method to polynomials in the complex plane and colour each point by the root it converges to. 6 presets, infinite zoom/pan, smooth shading, save PNG. Free.

Mathematics2DModerate60 FPS

⚙ Under the hood

This simulation allows users to explore the dynamics of the Newton fractal. It demonstrates how complex roots and basin boundaries emerge from iterative calculations, showcasing a fundamental aspect of chaos theory.

Newton FractalBasin BoundariesRoots

2D · HTML5 Canvas 2D · 60 FPS target · runs fully client-side, no install