Newton's Fractal | mysimulator.uk

Root-finding turned into a picture

Newton's method finds a root of a function by starting from a guess z₀ and repeatedly stepping toward wherever the function's tangent line crosses zero:

z_{n+1} = z_n − f(z_n) / f'(z_n)

For a real function this is a one-dimensional story: from a good starting guess the sequence converges to a nearby root, doubling its number of correct digits roughly every step (quadratic convergence). Newton's fractal comes from asking a more dangerous question: apply the exact same iteration with complex numbers, over the whole complex plane, to a polynomial like f(z) = zⁿ − 1, whose n roots are the n-th roots of unity spaced evenly around the unit circle. Colour every starting point z₀ by which root it eventually converges to, and shade the colour by how many iterations it took. The result is one of the most intricate fractals in mathematics, and it falls straight out of high-school calculus.

live demo · z³ − 1 = 0, basins of attraction coloured by root and speed● LIVE

Basins of attraction and why the boundary is a fractal

Near each root, Newton's method behaves exactly as you would expect: a whole neighbourhood of starting points converges quickly to that one root, forming a large, well-behaved basin of attraction. Far from the roots the story changes. Because f'(z) can be small or the correction step can overshoot wildly, points that are extremely close together can be sent to different roots, or take wildly different numbers of iterations to converge. The set of points that sit on the boundary between two or more basins — where arbitrarily small perturbations flip which root you land on — is exactly the set with this extreme sensitivity, and for Newton's method applied to a polynomial of degree three or higher, that boundary set is a fractal with self-similar structure at every scale: zoom into any boundary point and you find smaller copies of the whole pattern, an infinite regress of interlocking basins.

Why cubics and higher misbehave more than quadratics

For a quadratic like z² − 1 the two basins of attraction are simply the left and right half-planes, split by a straight line — no fractal behaviour at all. The trouble starts at degree three. For z³ − 1 the three basins interlock so intricately that every boundary point between any two basins is also a boundary point of the third basin — a striking topological fact called the Lakes of Wada property: three (or more) regions can share a common boundary everywhere, not just at isolated points. This is part of why the picture looks the way it does: colours from all three roots braid together no matter how far you zoom into any boundary region.

The connection to Julia sets

Newton's method for a polynomial f is itself a rational map z ↦ N(z) = z − f(z)/f'(z), and iterating any rational map on the complex plane is precisely the setting of complex dynamics that produces Julia sets and the Fatou/Julia dichotomy. The roots of f are superattracting fixed points of N (the derivative of N vanishes there, which is exactly why convergence is quadratic), the basins of attraction are the Fatou set of N, and the fractal boundary between basins is the Julia set of the map N. Newton's fractal is not merely like a Julia set — it literally is the Julia set of the Newton iteration function, which is why it shares the same self-similarity and boundary pathology as the Mandelbrot-adjacent fractals produced by iterating zⁿ + c.

Rendering it

The image is produced pixel by pixel: map each pixel to a complex number z₀, iterate the Newton step some fixed number of times (or until |f(z)| drops below a tolerance), and record which root the sequence ended nearest to and how many steps it took. Root identity picks the hue; iteration count picks the brightness, which is what carves the fine banding visible inside each basin, since points further from a root generally take more steps to converge. Points exactly on a Julia-set boundary technically never converge, so real implementations cap the iteration count and colour anything that hasn't converged by then with a distinct "non-convergent" shade, which is the dark webbing visible at the finest scale of the image.

Frequently asked questions

Why does Newton's method produce a fractal at all — isn't it just a calculus technique?

Newton's method is deterministic and converges cleanly for real-valued single-variable functions started near a root. Extended to the complex plane and iterated from every starting point, the extreme sensitivity of which root a point converges to — near the basin boundaries — produces a boundary set with fractal, self-similar structure, which is exactly the Julia set of the iteration map.

What is the Lakes of Wada property visible in the picture?

It is the surprising fact that for polynomials of degree three or higher, every boundary point between any two colour basins is simultaneously a boundary point of every other basin too — the regions share their entire boundary, not just isolated points, which is why the basins look interwoven at every zoom level rather than cleanly separated.

Does every point eventually converge to a root?

No. Points exactly on the fractal boundary (the Julia set of the Newton map) never settle on one root, and points very close to the boundary can take an enormous number of iterations. Any renderer has to cap the iteration count and colour the non-convergent points separately, which is the dark filigree visible in the image.

Try it live

Everything above runs in your browser — open Newton's Fractal and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.

▶ Open Newton's Fractal simulation

Newton's Fractal: When Root-Finding Draws a Fractal