Newton Fractal Explorer
Apply Newton's root-finding method to every point in the complex plane and color each pixel by the root it converges to. Zoom into the basin boundaries to find the impossibly intricate fractal between them.
Polynomial
Center re
0.00e+0
Center im
0.00e+0
Click the canvas to zoom in.
Newton's method in two dimensions
Newton's method is the classic numerical recipe for finding a root of a function: take a guess, follow the tangent line down to the x-axis, repeat. The update step is:
On the real line it converges quickly. Push it into the complex plane — let every pixel be a starting point — and the picture explodes. Every root of the polynomial has a basin of attraction: the set of starting points that converge to it. The boundaries between basins are infinitely fractured — the Newton fractal.
Reading the picture
- Hue: which root the pixel ended up at.
z³ − 1has 3 colours,z⁵ − 1has 5, etc. - Brightness: how fast it got there. Bright = quick (one or two steps); dark = slow (many iterations, near a boundary).
- Boundaries: three colours always meet at every boundary point. Between any two basins is always a third — that is the topological signature of Newton fractals.
- Relaxation a: setting
a ≠ 1over- or under-corrects each step, producing a relaxed family of variants (sometimes called "nova" fractals).
Cayley's paradox
In 1879 Arthur Cayley showed that Newton's method converges nicely for quadratics — the two basins are simple half-planes. He tried the same trick on the cubic z³ − 1 and... couldn't. The geometry of three-way basin boundaries was beyond 19th-century tools and the problem stayed open for nearly a century.
We can now see what defeated Cayley: any neighbourhood of a basin boundary contains points belonging to all three roots, fractally interleaved. The pathological polynomial z³ − 2z + 2 is even worse — Newton's method orbits 0 instead of converging, which is why a big region in that preset shows nearly-black noise.
Things to try
- Click a boundary to zoom in 1.8× and recentre — three colours always reappear.
- Slide relaxation up to 1.6 or down to 0.5 — the same polynomial grows new lobes and spirals.
- Compare orders: z³ vs z⁵ vs z⁸ at the same zoom — higher-order roots crowd together, producing tighter petals.
- Cayley's nightmare: open
z³ − 2z + 2and see why he gave up.