We will now associate a rational number **p/q** to each primary bulb.
The denominator **q** will simply be the period of the bulb, so we need
only describe how to find **p**. There are two ways to do this.

Given **c** in a primary bulb of period **q**, with **q > 1**,
first compute
**J _{c}** and then superimpose the attracting cycle of period

There is a second method to determine **p/q**, which is not quite well
defined but which, in practice, is easier to use. Again compute **J _{c}**
and the attracting cycle. As before, this cycle determines

The reason why this region is smallest is that it contains **F _{c}(0)**,
the image of the critical point of

The reason for the imprecision here is in the word "smallest." How
do we measure the size of these regions? It is in general impossible
to determine the areas of these regions explicitly. Thus in practice
we merely "eyeball" the various regions to see which is smallest.
Also, when determining **p** in this fashion, it is best to choose **c**
near the center of the bulb (as near as possible to the **c**-value for
which 0 lies on the attracting cycle). Of course, when **q** is large,
it is essentially impossible to distinguish the smallest region, so we
must resort to the previous method in these cases.

There is a third way to determine **p** that does not involve computing
**J _{c}
** but rather involves only looking at the antenna attached to the
bulb in

Now this is not always true if we measure the length using the usual
Euclidean distance. Rather, we should use a distance that assigns a
shorter length to the spokes closer to the main spoke. Without being
precise, this is what we mean by "hyberbolic eyeglasses." In Figure
6 we have displayed the **p/q** bulbs for various choices of **p** and
**q**. Click on the appropriate figure to view.

To test your ability to read off the rotation numbers, click here for a "clickable" version of Figure 6a-d which is in turn linked to enlargements and movies.

Fractal Geometry of the Mandelbrot Set (Cover Page)

3 Periods of the Bulbs (Previous Section)