Learning Your Way Around

Putting It All Together

Now that you understand the progression of periodicities and how to add periodicities together, you can find any periodicity you wish. Let's explore the decorations of Period 10 bulb, below.

Next we can apply the Spiral Rule. If we want to find a twisted version of this 10-armed pattern, just zoom down into the valley between the Period 10 bulb and the main body.

Verify that this structure still has 10 arms (though they're harder to count when they're so twisted).

If we zoom into the connections between any of the self-similar subunits in the picture above, we can apply the Bifurcation Rule, as illustrated in the sequence of four images below:

The rules you've learned about Spiraling, Bifurcation, and Periodicities can be applied anywhere in the entire Mandelbrot Set. For example, you could discover a 64-fold symmetric version of a highly twisted Period 23 object.

Fractivity:
Download the amazing and FREE open-source program XaoS which will allow you to explore the Mandelbrot Set (and other fractals) in more detail and with much more control.

Play with this program during the rest of this course. Explore the online tutorial and the many powerful features of the program. Don't be scared - it's easy to use - just point and click to zoom in!

Your term project is to create the most interesting and beautiful fractal you can. Submit both the image and the fractal file by email by the end of the term.

In the next chapter, we will learn how an extremely simple equation creates the Mandelbrot Set...