Newton’s Method

Three Roots, Three Basins

The Step, and Why It Pulls So Hard

Each press of the solver replaces every guess with the point where the tangent line crosses zero. For this polynomial that works out to one line:

Close to a root the pull is extreme. The distance to the root is roughly squared at every step, so the number of correct digits about doubles each time. For the three cube roots of one the squaring is exact in the limit: the constant in front, |f″/2f′| at the root, equals 1 at all three. Here is one guess, started 0.3 from the root at 1 and followed step by step:

The ratio settles on 1, as the formula says it should. One step later the guess sits on the root to the precision the computer can hold.

The Basin, Measured

The card says a nudged guess comes back. Nudge the guesses pushes every guess a random distance between 0.08 and 0.38 in a random direction. That was tested 200,000 times at each of the three roots, starting from the root itself:

Every one came back. The reason is how far the edge of each basin is from its root. Searching every direction from the root at 1, the nearest point belonging to another basin is 0.623 away, in two directions about 138° either side of straight out. The panel’s largest nudge is 0.38, well inside that. A push harder than 0.623 in the wrong direction would send a guess to a different root, and the three roots are symmetric, so the same distance holds at all three.

Across the whole picture, the root at 1 takes 35.3% of the square and the other two take 32.4% each. That is the square frame, not the method: measured on a disc centred on zero, each root takes one third to within 0.01%.

Press Stop the solver and the basin is gone. No guess moves, near a root or far from one, and a nudged guess stays exactly where the nudge left it.

Where Two Colours Meet, the Third Is There

The basins do not meet along clean lines. A Yale course page on Newton’s method states the property directly: “any circle enclosing points of two of these colors must also enclose points of the third color. This is called the Wada property.”

The panel’s own map was checked for it. Two thousand points were picked at random along the edges between colours in the background, and a small disc around each was sampled to see which roots its points reach. The discs were then shrunk, a factor of ten at a time:

A first pass of 1,024 samples per disc missed the third colour in 18, 9 and 1 of those discs at the three smaller sizes. Sampled 32 times more densely, every one of them turned it up. Fewer discs catch two colours as they shrink because a random point near the edge is less and less likely to sit on it. At every scale tested, no disc was found with only two colours. That is consistent with the published property. It is not a proof of it; the proof belongs to the mathematics, not to the panel.

Why This Is an Attractlet

The kernel sorts attractlets into kinds. One is the Attractlet Model (Formal or Symbolic): a formal, mathematical, or symbolic representation that behaves like an attractor within its formalism, but whose persistence is “sustained by the computational, mathematical, or symbolic substrate,” which is itself sustained by sovereign attractors: the mathematicians, computers, or formal systems running the model. The kernel counts numerical solvers in this kind by name (ATTRACTLET, § 7.4).

Newton’s method is a numerical solver. Its roots do not hold the guesses; the next application of the formula moves them. Nothing in the picture produces that application. Take it away and there is no pull left to find.

The fractal boundary does not move the classification either. § 7.4 states it directly: “No attractlet may be promoted to attractor status by complexity, longevity, apparent stability, mathematical elegance, or external coupling strength.” Everything measured above, the squaring, the return from every nudge, the third colour at every scale, is a property of the formula. A basin shows that a structure returns. It cannot show where the return comes from.

What This Panel Is Not

One polynomial. Everything here is for z³ − 1. Other polynomials give other pictures, and nothing measured here is claimed for them.

The full step, never a partial one. The panel always moves a guess the whole way to where the tangent crosses zero. Shortening the step changes the basins and slows the squaring.

“Settled” is a threshold. A guess counts as settled within 0.001 of a root. The background stops after 40 steps; at that limit 6 of its 518,400 pixels have not yet settled, and all of them do within 200.

Zero has no step. The tangent is flat at z = 0, so the formula is undefined there. The panel leaves a guess at exactly zero where it is.

The motion between steps is drawn for the eye. Each guess changes once per step. The glide between is a display effect and is not part of the model.

The boundary check samples; it does not prove. Four sizes and a few thousand discs cannot rule out a two-colour disc somewhere smaller or somewhere unsampled, and ordinary computer arithmetic runs out well before the boundary does.

“Julia Sets and the Mandelbrot Set,” Fractal Geometry, Yale University, users.math.yale.edu/public_html/People/frame/Fractals/MandelSet/ComplexNewton/NewtonBasins/Basins3.html (fetched and read 2026-09-16). Source of the quoted Wada property.
“Lecture 29: The Newton-Raphson method as a dynamical system on R,” University of Waterloo course notes, links.uwaterloo.ca/amath731docs/newton_method_complex_plane_PMATH350.pdf (fetched and read 2026-09-16). Source of the quadratic-convergence bound and the shared boundary of the three basins.
Every number on this page was measured on the panel’s own rule, with its own settling threshold, on 2026-09-16.