Solve z³ = 1 by Newton’s method: why do fractals grow between the roots’ territories?

To solve $f(x)=0$, guess $x_0$, draw the tangent to the curve there, and take where it meets the axis as a better guess:

Finding $\sqrt2$ means solving $x^2-2=0$. From $x_0=1$: $1.5$, $1.41667$, $1.4142157$, $1.41421356237469$, with 1, 2, 5 and 11 correct decimal places. Near a root the error is roughly squared each step and the digits double: quadratic convergence.

The same formula works for complex numbers, with the complex derivative as the "tangent". For $p(z)=(z-r_1)(z-r_2)\cdots(z-r_n)$ one Newton step can be written with the roots alone:

That is how the stage computes it, which is why you can drag the roots.

For $z^2-1$ the points equally far from $\pm1$ form the imaginary axis; starts on either side go to the nearer root, and the border is a straight line, as Cayley proved in 1879. Of $z^3-1$ he wrote that the case "appears to present considerable difficulty".

For $z^3-1$ one Newton step is $z\to\dfrac{2z^3+1}{3z^2}$. At $z=0$ the derivative is 0, the tangent is flat and the step divides by zero; points near 0 are flung far away in one step.

The point $-2^{-1/3}\approx-0.7937$ on the real axis lands exactly on 0 in one step. Its neighbours land all around 0 and are then sent in every direction, so all three colours crowd around it. Infinitely many points land on points like this one, and around each the same thing happens: petals on petals.

A theorem of complex dynamics says that for maps like this the border of every territory is one and the same set (the Julia set). So every border point touches all three territories, the "three-basin property", the same strange phenomenon as the Lakes of Wada in topology.

For $p(z)=z^3-2z+2$ one Newton step sends 0 to 1 and 1 back to 0:

And since $p''(0)=0$ the cycle pulls nearby points in, so a whole patch of starts around 0 and around 1 never reaches a root, shown in black. In practice equation solvers check for convergence and try another start when it fails.