(All images created with MATLAB® software)
Figure 1: Newton's method of attraction basins for z3= 1.
Basins of attraction for the three cubic roots of unity under iterations using Newton's method. Each color in Figure 1 corresponds to a root. The complicated boundary between the regions has a self-similar fractal structure: any piece of it that is blown up will display the whole pattern.
Picture obtained using the script NewtonRoots in the Athena 18.04 MATLAB® Toolkit.
Figure 1: Newton's method of attraction basins for z3=1.