Experiment № 050 · The story
Basins
Why I picked it
Most fractals I've built for this collection were invented to be looked at. This one wasn't. Newton's method is a piece of plumbing — the thing your calculator quietly does when you ask it to solve something — and it only turned into a picture because someone asked an innocent bookkeeping question: if I start Newton's method here, which root do I end up at? Arthur Cayley asked exactly that in 1879. For a quadratic he solved it in an afternoon: the plane splits along a straight line, each half falls to its nearer root, done. Then he tried the cubic and wrote that the case "appears to present considerable difficulty" — and stopped. He was right to stop. The answer needed another hundred years and a computer, and it is this: the three territories interlock along a boundary that has no smooth piece anywhere. I wanted experiment 050 to be the one where a boring numerical routine turns out to have been hiding a monster.
What I enjoy about it
The hover trace, more than the picture. Move the cursor and you see the actual orbit — the little chain of hops Newton takes from where you pointed to whichever root wins — and out in the flat interior of a basin it's three confident jumps and done, while a pixel or two away, on the frilly border, the same algorithm goes skittering across the plane for forty steps before it makes up its mind. Same rule, same polynomial, neighbouring starting points. I also love that the page doesn't get told where the roots are: it finds them itself with Durand–Kerner before it draws anything, then colours by whichever one each pixel surrendered to. And do try z³ − 2z + 2, the last preset — those black lagoons are the famous embarrassment, whole open regions where Newton's method locks into a two-step cycle and bounces forever, never converging to anything. The twist slider (which rotates the correction step into the complex plane) is the one control that isn't textbook: it turns the basins into pinwheels and makes it very obvious that these shapes are about the dynamics, not the polynomial.
Something you might take away
Zoom into any point on the boundary and you'll never find a stretch where just two colours meet. Every single boundary point touches all the basins at once — mathematicians call it the Wada property, after a set of lakes in a Japanese thought experiment that all share one shoreline. That's the real reason the edge is fractal: with three or more equal claimants there is no way to draw a fair frontier between two of them without the third one turning up as well, at every scale, forever. Whenever a boundary has to separate three things that all have an equal claim, "just draw the line" quietly stops being an option.