Newton Fractal — z³ − 1
Applying Newton's root-finding method to z³ = 1 in the complex plane reveals three fractal basins of attraction.
Newton’s Method in the Complex Plane
Newton’s method finds roots of a function by iterating:
z_{n+1} = z_n − f(z_n) / f′(z_n)
For f(z) = z³ − 1, this becomes:
z_{n+1} = (2z_n³ + 1) / (3z_n²)
The equation z³ = 1 has three roots in the complex plane, spaced 120° apart on the unit circle.
The three roots marked on the complex plane: 1, −½ + i√3/2, and −½ − i√3/2. Starting from any initial guess z₀, Newton’s method should converge to one of these three roots.
The Fractal Boundary
From most starting points, Newton’s method converges quickly to the nearest root. But the boundary between the three basins of attraction is not a smooth curve — it is a fractal of infinite complexity.
Each pixel is coloured by which root Newton’s method converges to from that starting point (blue, green, red), with brightness proportional to the speed of convergence. The three regions have a threefold rotational symmetry — but at every scale their shared boundary is a fractal curve, with each region’s boundary touching the other two.