Cubic equation: f(z) = z³+az+b

    Since a and b are complex, each having 2 real components.  Therefore the cubic above has 4 real dimensions (a=a_r+a_ii and b=b_r+b_ii).  In order to explore the set, we have to take cross-sections through the set, holding at least one of the 4 real values constant.  If one real variable is held constant, we can explore the set in 3 dimensions.  The representation of these 3 dimensions at once becomes problematic, so here I have represented the set as 2-dimensional cross sections, either holding a constant or b constant and displaying the set cutting through the b-plane or a-plane, respectively.

As with the traditional Mandelbrot set, we have to choose an initial point to begin the complex iteration.  The following theorem by Fatou indicates the point at which we should start:  If there is a stable cycle, a critical point will iterate to it.

In the 2-dimensional set, we start with z=0, the critical point of the equation z²+c.  The critical points of f, the cubic above, are +sqrt(-a/3) and -sqrt(-a/3).

Most of the images in the Surveys and Details sections were produced with 250 iterations of the above equation.