Mandelscott: Mandelbrot on Steroids

If you've read (and remembered) my earlier posts, the Mandelbrot set is generated by iterating the complex number z with the equation z^2 + c. Have you ever wondered what effect the power of two has on the set? In other words, what if we pumped up the Mandelbrot and iterated z^3 + c?

You would find what is called the Cubic Mandelbrot. (The original Mandelbrot is considered Quadratic.) Picture please!

You can see the some parts of the Quadratic Mandelbrot and doubled in the Cubic Mandelbrot. I like how the bulbs on the top and bottom (and smaller bulb clones on the sides) look like insect antennae.

OK, let's zoom in:

The "antennae" continue on forever, and have little Cubic Mandelbrots in them! To wit:

Let's look at some close ups with different coloring:

This last one looks a little bent from the original. I'm not sure what causes it, but I have seen distortions in other fractals like that when zooming up close.

Here are a few more of the Cubic Mandelbrot before I move on. I like how these look like tropical islands. Note the self-similarity at varying scales even at this extreme close-up.

Just as there's a Cubic Mandelbrot, there is also a Cubic Mandelbar. Some examples:

Interestingly, while the Cubic Mandelbrot has two bulbs, the Cubic Mandelbar has four arms. But you should also notice the "antennae" at the end of each arm. Zooming in on one (and changing the color scheme):

Up close, it looks a lot like the Cubic Mandelbrot (except for the tiny Cubic Mandelbars on the antennae).

Zooming in even more,

we find tiny Cubic Mandelbrots inside the Cubic Mandelbar. Considering the same thing happens in the Quadratic Mandelbrot/Mandelbar, this is not surprising. But it remains fascinating.