Exploration #6

Question: What is the period of the "primary bulbs" and how can you see this period? For that matter, what are the primary bulbs?

Answer: First of all, the "primary bulbs" are the large, disk-like decorations hanging off the main cardioid in the Mandelbrot set. To the right is a picture. The largest black region in the Mandelbrot set is the "main cardioid." Off it hang infinitely many decorations, each of which feature a large bulb from which dangle infinitely many other smaller bulbs together with complicated antenna-like structures. We'll have fun with the antennas later. For now, though, it is important to remember that a "primary bulb" is that large disk-like region directly attached to the main cardioid, not all the other junk that hangs from it. Those are the "secondary bulbs" and antennas that we'll visit later. Incidentally, the picture shows a magnification of a period 5 bulb together with its location in the Mandelbrot set.

Exploration: In the Mandelbrot set below, select one of the white dots. These correspond to specific c-values in one of the primary bulbs. You will then see the corresponding filled Julia set together with the fate of the orbit of 0 as well as a magnification of the primary bulb from which the particular c-value was selected. Note that the orbit of 0 tends to a cycle of some period, and that the period changes with the bulb. This period is called the period of the bulb. If you select a different bulb, you (sometimes) see a different period.

You can see the period of the bulb by simply counting how many iterations it takes for the orbit displayed to begin repeating itself. However, there are two other ways that you can read off this period, one involving the geometry of the filled Julia set, the other involving the geometry of the Mandelbrot set. Your job is to figure out how to read off these periods just from the geometric information. You should be able to look at a filled Julia set drawn from a primary bulb and read off the period of that bulb. You should also be able to look at a primary bulb in the Mandelbrot set and also read off the period.

Further Exploration: To choose other c-values, you may use the Mandelbrot/Julia Set Applet. This tool will take some time to compute various Julia sets, since, unlike the above images, you really compute each different image.

Now answer the following questions:

Question 6A:

Here is a filled Julia set
that comes from a c-value
inside a primary bulb
in the Mandelbrot set.
What is the period of this bulb?


Question 6B:

Here is a primary bulb
in the Mandelbrot set.
What is the period of this bulb?


For Further Information: