Lab X · 解収束

Every point chooses
a solution.

Newton's method asks one thing: from where you are, which way is downhill on |p(z)|? Take that step, repeat. For almost every starting point in the complex plane, the iteration converges to one of the three roots of p(z) = z³ − 1.

Each pixel here is coloured by which root it lands on; brightness shows how quickly. The delicate filigree between regions is the Julia set of the iteration — and at every point on it, all three colours meet. There is no smooth way to partition the plane into three solution-basins.

We are made of solutionists. The trick is knowing which root you are walking toward.