The reason you iterate z = 0 with the quadratic is that it's a critical point of f(z) = z^2 + 1. By analogy, for the cubic you should iterate the critical points (plural) and (I hope I don't screw this up!) see if *at least one* escapes. (If I'm wrong, then you see if *both* escape; unfortunately, I know nothing about dynamics. As I said, I don't work in the area!) The derivative for the cubic is 3z^2 + (c - 1). So you need to set this equal to 0 and solve for z (which gives two complex square roots, which gives two critical points). You iterate these guys, not z = 0. Thus, you'll be iterating different points, depending on the current value of c.

In these kinds of situations I have my computer study the fate of both points, and plot distinct color gradients depending on what pattern occurs. The results can be quite spectacular. With Ultra Fractal there's a nifty trick. Make a formula, and make the critical point selection a parameter. Generate a two-layer image with each layer having the same settings save critical point selection. Give them distinct outside gradients. Then you can get, say, black where both points are trapped, dark gradients where one escapes and a different shade depending on which, and where both escape, a colorful gradient that has filaments of two different colors present. Inside Mandelbrot-like regions can be found dendrites studded with filled-in Julia set like forms, as well as more mini Mandelbrots, in black; the Julia like forms have both critical points sucked into the same attractor, and the smaller Mandelbrot figures have them going to separate attractors. Of course, it gets hairier with quartics. Then there are three critical points...

Feel free to copy this to others if it helps to correct this. Oh ... do any of those fractal-generating programs actually do the cubic, but the *correct* way? If not, you could do a favor for the people who write these things by letting them know.

Well I have both Fractint formulas, Ultra Fractal formulas, and C code that produce generic cubic images. They use the two-parameter cubic family z^3 - 3a^2z + c, however, with critical points +/-a.

(Maybe go ask a dynamical systems person first whether you want *at least one* or whether you want *both* critical points to escape in order to color the point c. Don't take my word for it --- as you can see, I'm not a good source for this!)

This depends on what kind of picture you want to generate. The usual rigorous Mandelbrot analogue is the set where all critical points are trapped. For all polynomials, this set is connected, though you may be looking at a space with a large number of parameters. For each critical point you can get interesting fate maps. To really have a map of the full dynamic behavior of the function, though, you need to do the Ultra Fractal layering thing. (In particular, the high-iteration stuff of different critical points should get distinct colors, which means layering with different gradients.)Get more from the Web. FREE MSN Explorer download : http://explorer.msn.com