Thought Toys · Chaos & fractals · Exhibit 34
Newton's method is the textbook trick for finding a root: from a guess, follow the tangent line down to where it crosses zero, and repeat. Here there are three roots, so every starting point is really a vote for one of three answers. Colour the whole plane by which answer each point votes for, and the borders between the votes turn out not to be lines at all.
Choose a starting point
x = 0.750 · converges to root A after 8 steps
The equation is as simple as cubic equations get: z³ = 1. In the complex plane it has exactly three solutions, spaced like the hours 12, 4, and 8 on a clock face. Newton's method — the calculus-101 way to find a root: at your current guess, follow the tangent line down to where it hits zero, and use that as the next guess — will, from almost any starting point, home in on one of the three. Colour every point in the plane by which of the three it eventually lands on, and you get the picture behind the scan line: three broad, calm regions (the basins of attraction), each clearly the turf of one root.
Drag the starting point slowly through one of those regions and nothing surprising happens — root A stays root A, root A stays root A, for a long, boring stretch. But approach a border and something strange takes over. The colours don't switch cleanly from amber to cyan at one point. Instead they interleave — amber, cyan, green, amber, cyan — in a spray of ever-finer slivers, and however far you zoom into any of it, the same three-colour braid keeps reappearing. That braid is the boundary; it has no smooth edge to finally cross. (The detail slider only controls how long the background render is allowed to keep iterating each pixel before giving up — a low setting can leave the finest slivers unresolved and dim, the same honest trade-off as the cabinet's Mandelbrot exhibit. The white orbit trail and its readout below always run the real rule to full convergence, independent of that setting.)
Here is the part worth sitting with. This whole picture is completely deterministic — the same starting point always lands on the same root, with no randomness anywhere in the rule. And yet, drag the slider one notch at a time through a border region and the "step once" readout below will sometimes flip roots between two starting points so close together that no ruler, and eventually no floating-point number, could tell them apart. A perfectly precise, perfectly repeatable rule can still be practically unpredictable, just by living too close to one of these borders.
improve/verify/34-newton-fractal.js):
typical points converge exactly onto a root (residual |z³−1| under 1e-9); a scan across the boundary flips
root 13 times, not once; bisection finds two starting points 5.7×10⁻¹⁶ apart — as close as two adjacent
floating-point numbers can be — that still converge to different roots. Counter-examples: a
whole neighbourhood of points deep inside a basin (near root A) all agree on the same answer — the fractal
lives only on the border, basins have solid interiors — and starting exactly at z=0, where
f′(z)=0, the method divides by zero and never converges at all: not every point even has a fate.
Also in Chaos & fractals: Two nearly identical starts, torn apart →
← the cabinet · Thought Toys — a cabinet of explorable explanations. Exhibit 34.