Newton Basins
w = z −(z³ − 1)/(3z²)
Open in the app The dials and keys named below are the app's.
What it draws
w = z − (z³ − 1)/(3z²) is one step of Newton's method for the equation z³ = 1: subtract the value of z³ − 1 divided by its derivative 3z². With Iterations at 24 the step is taken twenty-four times from every point of the square, which Span of z sets to ±2. The colour of a point says where its run ended.
The three answers are the cube roots of one: 1, −0.5 + 0.866i and −0.5 − 0.866i, spaced 120° apart on the unit circle. Each is an attracting fixed point of the step, and the convergence is quadratic. Twenty-four steps are far more than enough: almost every starting point lands on a root to full precision long before the count runs out. Hue is the argument of the number reached, so each root paints its basin one flat hue, and the three hues stand 120° apart on the colour wheel.
The pit at the origin
Nothing that converges ever escapes, so the converged sheet is a plateau of one height. The relief comes from the one place the step misbehaves: it divides by 3z². A point within 1/√12 ≈ 0.289 of the origin is flung far enough in a single step to count as escaped, so it takes the outside tone with no hue and drops well below the plateau. That is the pit in the middle. Every point that lands in that disc at a later step falls too, so smaller copies of the pit speckle the picture. Over this square about one point in thirteen escapes within the twenty-four steps.
Why the boundary is a fractal
Where two basins meet, the third is there as well. Every boundary point has points of all three basins arbitrarily close to it, so no smooth curve can separate them and the boundary is forced to be infinitely detailed. It is the Julia set of the Newton step. The practical moral is that Newton's method is exquisitely sensitive to its start near that boundary, however tame the equation.
Try
- Top for the flat picture, three hues pinwheeling about the origin.
- Solve a quartic instead: z − (z⁴ − 1)/(4z³) gives four basins, for 1, i, −1 and −i.
- Drop Iterations to 4: many points have not settled yet, and the basins are blurred bands rather than flat colour.
- Lower Span of z to 0.6 to fill the view with the pit at the origin and the boundary wrapped round it.
- Raise Height: the plateau and the pits deepen together, and the sheet reads as a table with holes punched in it.