When root-finding algorithms produce infinite complexity
Newton's method is a root-finding algorithm: given a starting point, it iteratively refines guesses to find where f(z) = 0. For real numbers, this works beautifully. But extend it to the complex plane, and something magical happens at the boundaries...
Newton's method is completely deterministic—the same starting point always produces the same result. Yet the boundaries between basins of attraction are infinitely complex fractals. A tiny change in starting position can lead to converging to a completely different root. This is chaos emerging from simple rules!
For z³-1, every point on the fractal boundary is simultaneously on the boundary of ALL THREE basins. No two colors ever share a simple border without the third squeezing in between. This "Wada property" was discovered by mathematician Kunizō Yoneyama in 1917.
Isaac Newton developed his method around 1670. Arthur Cayley first studied basins of attraction in the complex plane in 1879, famously writing he could make no progress on the z³-1 case. The fractal nature wasn't understood until the advent of computer graphics in the 1980s.