Fractal Geometry of the Mandelbrot Set (Cover Page)

2 The Mandelbrot Set (Previous Section)

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Note that the Mandelbrot set consists of many small decorations. Closer inspection of these decorations shows that all of them are different in shape.

** Figure 4. Several decorations on the Mandelbrot set**

For example, consider any decoration directly attached to the main
cardioid in **M**. We call this bulb a * primary * bulb or
decoration. This decoration in turn has infinitely many smaller
decorations attached, as well as what appear to be antennas. In
particular, as is clearly visible in Figure 4, the "main antenna"
attached to each decoration seems to consist of a number of spokes
which varies from decoration to decoration.

There is a beautiful relationship between the number of spokes on
these antennas and the dynamics of **x ^{2} + c**
for

At this point, it is useful to perform a series of computer experiments that yield the periods of some of the other primary decorations. You can easily check the following facts using a computer and any of a number of available software packages. (See, for example [9] or [12].)

It is easy to check that ** M ** is symmetric about the real axis
and that these periods hold for the complex conjugate **c**-values. This
gives us a larger table of results. In Figure 5 we have summarized
some of these results graphically.

** Figure 5. Periods of the primary bulbs in M **

Further experimentation shows that there is a remarkable relationship between the number of spokes in the largest antenna attached to a primary decoration and the period of that decoration. These numbers are exactly the same! (Don't forget to count the spoke emanating from the primary decoration to the main junction point.) See Figure 6.

* Period 3 Bulb*

* Period 4 Bulb*

* Period 5 Bulb*

* Period 7 Bulb*

** Figure 6. Note that the period of the bulb is the
same**
** as the number of spokes in the antenna**

The following short quiz will help you develop your skills at recognizing the period of a bulb on the Mandelbrot Set.

Fractal Geometry of the Mandelbrot Set (Cover Page)

2 The Mandelbrot Set (Previous Section)

BU Math Home Page

Prof. Robert L. Devaney (Boston University)