Now let's use the rotation numbers to investigate the arrangement of
the bulbs around the main cardioid. This is one of the main
experiments performed in the chaos club. The idea is to use the
computer to determine experimentally the location of the **p/q** bulbs
for **q** between 2 and 7. A number of these are sketched in Figure 7. Note
that the bulbs are ordered exactly as the rational numbers! The fact
is that there is a unique **p/q** bulb for each rational between 0 and 1
and these fractions are ordered in the natural way around the cardioid
(beginning at 0 at the cusp and proceeding in a counterclockwise
direction around the boundary ending at 1, again at the cusp). That
is, the Mandelbrot set serves as a real number line (or at least a
rational number line). Moreover, unlike the real line, each number on
**M** can be clearly identified by looking at the antennas and
counting the spokes.

Figure 7

The reason why this is true demands a bit of complex calculus. Recall
that the fixed points of **F _{c}** are given by the roots of

We have

Along the boundary of the main cardioid, we saw earlier that

As increases from 0 to
2 Pi, **c** travels once around the
boundary of the cardioid in the counterclockwise direction. It follows
that the fixed points are given by

when **c** lies on the boundary. Note that

so that is a * neutral* fixed
point. If we were to compute the value of at the point where
the **p/q** bulb attaches to the main cardioid, then it is a fact that

What happens is a fixed point and a period **q**
point coalesce into a neutral fixed point whose derivative is

Figure 8. A movie of the attracting cycle in the 1/3 bulb.

Fractal Geometry of the Mandelbrot Set (Cover Page)

4 Rotation Numbers (Previous Section)