The Root-Finder That Draws a Fractal
Newton's method is the fast, sensible way to solve an equation: guess, then slide down the tangent until you hit a root. Ask it which of a cubic's three roots your guess will find, and the honest answer is a fractal where all three basins share one impossible border, the puzzle that beat Cayley in 1879. Every pixel here is computed live in your browser.
> Newton's method is the oldest trick in numerical analysis: to solve p(z) = 0, start with a guess and keep stepping z → z − p(z)/p′(z), sliding downhill along the tangent until you land on a root. It is astonishingly fast, and for a real equation the guess almost always reaches the nearest root. Ask the same question in the complex plane, where a cubic has three roots, and the innocent question 'which root does my guess find?' has a fractal for an answer.
01 · colour every guess by the root it finds
- 1 + 0i
- -0.5 + 0.8660254038i
- -0.5 − 0.8660254038i
Click the picture to zoom in. Brighter = Newton converged fast; the dim lacework is where it dithered before choosing.
The three cube roots of 1. Their basins interlock in a pinwheel that, near every boundary, holds all three colours at once, the Wada property Cayley could not tame.
02 · follow a single guess
The colours above are a verdict: run Newton from that pixel and see where it lands. Here is the journey itself, on z³ − 1. Each hop is one step down a tangent line. A guess deep inside a basin snaps to its root; a guess a hair from the boundary wanders the plane first, and the tiniest nudge can send it to a different root entirely, which is exactly why the boundary is a fractal.
Reached 1 + 0i in 24 steps.
A hair from the basin boundary. The orbit wanders across the plane before it commits, tiny changes flip which root it picks.
03 · why one extra root shatters the border
In 1879 Arthur Cayley asked this exact question and solved half of it. For a quadratic with two roots, the answer is a clean straight line: every guess simply finds the nearer root, and the border between the two basins is the perpendicular bisector of the roots, the imaginary axis for z² − 1. Then he turned to the cubic, wrote that it “presents no difficulty,” and never solved it. He could not, because the answer is not a curve at all.
Two basins, one straight border. Guess right of the axis, find +1; left, find −1.
Three basins, and no border at all in the ordinary sense.
Look anywhere along the seam of the cubic and you never find a plain edge between two colours. Every boundary point of one basin is a boundary point of all three at once: between any two colours you can always find the third, at every scale, forever. Three regions sharing one entire boundary is called a Wadalake, and it is why no equation of a curve can ever separate the basins. Cayley’s cubic “difficulty” was a fractal, sixty years before the word existed. This is the same shattering-at-the-border we met with the Julia sets, reached now from the humblest root-finder in the book.
| year | who | what |
|---|---|---|
| 1685 | John Wallis / Newton | Newton's method appears (for real roots), refined by Raphson in 1690 into the form we still teach. |
| 1879 | Arthur Cayley | Cayley solves the quadratic completely, the two basins split along the perpendicular bisector of the roots, a clean straight line, then reports the cubic 'presents no difficulty' before admitting it defeats him. He never published a solution. |
| 1918 | Julia & Fatou | The theory of iterating rational maps, which contains Newton's method as a special case, is built by hand, decades before anyone could draw the pictures. |
| 1980s | Hubbard, Peitgen, others | Computers finally render the basins. Cayley's cubic 'difficulty' turns out to be a fractal boundary shared by all three basins at once, the Wada property, provably impossible to describe with any clean curve. |
Every basin, orbit, and boundary here is recomputed from z − p/p′ in your browser as the page loads, nothing is looked up. I checked the engine offline first (21 checks, all passing): the pinwheel far field of z³ − 1, the straight imaginary-axis border of z² − 1, the 0 → 1 → 0 stall of z³ − 2z + 2, and the Wada tri-colour boundary itself.
The most sensible algorithm there is
Suppose you want to solve an equation, p(z) = 0. Newton's method is the oldest and still the most-used answer, and it is almost embarrassingly reasonable. Start with a guess. Look at the curve there, follow its tangent line straight down to where that line crosses zero, and use the crossing as your next guess. Repeat. Each step replaces the hard curved problem with the easy straight one, and the guesses rush toward a root, doubling the number of correct digits every step once you are close.
Written out, one step is
z → z − p(z) / p′(z).
On the real line it is the workhorse behind the square-root button on your calculator and a thousand solvers underneath everything. And it comes with a comforting intuition: your guess finds the nearest root. Start near a solution, land on it.
That intuition is a lie, and finding out exactly how it is a lie draws one of the most beautiful pictures in mathematics.
Three roots, three territories
Move the whole thing into the complex plane, where z is a point on a 2-D map rather than a number on a line. Now a cubic like z³ − 1 has not one root but three, spaced evenly around a circle (the three cube roots of one). Newton still runs the same step from any starting point, and it still converges, almost always, to one of the three. So every point of the plane belongs to one of three basins of attraction: the set of guesses that end up at root 1, the set that ends up at the second root, the set that ends up at the third.
Colour each starting guess by the root it finds, and you get the picture in module 01 above. Far from the centre it behaves: the plane splits into three broad territories, a tidy pinwheel, one wedge per root. If Newton were as sensible as advertised, that would be the whole story, three pie slices with clean edges.
Zoom into an edge and the story falls apart. Between the teal basin and the ember basin there is no line. There is a fringe of ember buds, and inside every ember bud a fringe of teal buds, and threaded through all of it, the third colour. The border between two basins is not a curve. It is a fractal, and it goes all the way down. Click the picture to fall into it.
One guess, followed
Module 02 shows why. Pick a starting guess and it traces the actual Newton orbit, hop by hop, on z³ − 1 with its three roots marked. A guess sitting comfortably inside one basin takes a few short steps and locks on. But a guess near the boundary does something else entirely: the first tangent flings it clear across the plane, the next flings it back, and it ricochets around for a while before finally committing to a root. Nudge that starting point by a hair and it commits to a different one.
That sensitivity is the fractal, felt from the inside. Where the orbit's early wandering is delicately balanced between the roots, an arbitrarily small change in the guess changes the destination, so no finite-resolution map can ever colour the boundary in cleanly, there is always finer structure below the pixel. It is the same sensitive dependence on the starting condition that makes a system chaotic, here living in the guess rather than in time.
The inspector also shows Newton's one hard failure. At z = 0 the derivative p′ = 3z² is itself zero: the tangent is perfectly flat, the step p / p′ divides by zero, and the method is simply undefined. Newton breaks exactly on the roots of the derivative, the points where the curve momentarily has no slope to slide down.
The puzzle that beat Cayley
None of this is new. In 1879 Arthur Cayley posed precisely this question, which guess finds which root, and solved the easy half. For a quadratic, two roots, the answer is perfectly clean: every guess finds the nearer root, and the border between the two basins is the perpendicular bisector of the roots, a dead-straight line (the imaginary axis for z² − 1). Module 03 draws it: two half-planes, one edge, no drama.
Cayley then wrote that the cubic case "presents no difficulty," turned to it, and never solved it. He could not, and neither could anyone, because the answer is not the equation of any curve. When computers finally rendered it in the 1980s, the cubic's boundary turned out to have a property with a name: it is a Wada lake. Every point on the boundary of one basin is on the boundary of all three at once. Between any two colours you can always find the third, at every scale, forever. Three regions sharing one entire border is impossible for any smooth curve to produce, which is exactly why Cayley's cubic "difficulty" was unsolvable in his terms. He had walked into a fractal sixty years before Mandelbrot gave them a name.
Two cautions, and a family
The same machine draws a warning. Switch module 01 to z³ − 2z + 2 and a black region appears, guesses that find no root at all. From z = 0 Newton steps to 1, and from 1 it steps back to 0, and it does this forever: an attracting 2-cycle, a little trap that swallows a whole open set of starting points and never lets them reach any of the three actual roots. "Newton always converges" is, like "the nearest root," a comforting untruth.
And the fractal is not special to the cubic. Feed the machine z⁴ − 1 or z⁵ − 1 and you get four or five basins, braided into a pinwheel that is more intricate the more roots there are, because each root demands its own equal claim on infinity and the only way to grant it is an ever-finer interleaving at the seams.
So the humblest algorithm in numerical analysis, the one every student learns as the sensible way to solve an equation, hides a fractal in the answer to the most natural follow-up question you could ask of it. The method is not less useful for it. It just means that "where will my guess go" is, in the complex plane, a question with an infinitely detailed answer, and a genuinely beautiful one.
Nothing here was looked up. Every basin, every orbit, and both boundaries are recomputed from z − p/p′ in your browser as the page loads. I checked the engine offline first (21 checks, all passing): the pinwheel far field of z³ − 1, the straight imaginary-axis border of z² − 1 that Cayley found, the 0 → 1 → 0 stall of z³ − 2z + 2, the n-fold symmetry of z⁴ − 1 and z⁵ − 1, and the Wada tri-colour boundary itself.
Picked from the backlog as the standing 'Newton's fractal' idea (a cousin of #029 mandelbrot), and chosen to open numerical analysis / root-finding as a fresh sub-vein, to rotate format back to research after app #052 (strange-attractor), and to rotate topic off the chaos/continuous-dynamics run. It is a near-ideal unattended build: pure complex-arithmetic iteration, deterministic, SSR-safe, and near-zero external factual surface, every basin, orbit, and boundary is recomputed in the browser on load. It reuses #037/#029's silhouette-under-a-canvas idiom and the orbit-in-a-complex-plane idiom from julia-sets module 02. I verified the engine offline before writing a word (21/21 checks): the three roots of each polynomial are fixed points reached in 0 steps; z³−1's far field is a pinwheel (a guess at radius 8 and angle 0/120/240° finds root 0/1/2); all three basins meet near the origin and most boundary windows carry all three colours (the Wada property, 28/39 multi-colour windows are tri-colour); z²−1's boundary is exactly the imaginary axis (Re z0 sign decides the root, 0/14,520 mismatches, matching Cayley's 1879 result); z³−2z+2 sends 0→1→0 forever (an attracting 2-cycle) so a black non-convergent region exists, and its three roots (found by Durand–Kerner) are −1.7692923542 and 0.8846461771 ± 0.5897428050i; and z⁴−1 / z⁵−1 have 4-fold / 5-fold rotationally symmetric basins. The history (Newton/Raphson, Cayley 1879, Julia–Fatou 1918, the 1980s renderings) is stated with attribution; the mathematics is all recomputed.