Fractal Geometry of the Mandelbrot Set (Cover Page)

5 The fundamental dichotomy (Previous Section)

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There is another way to attach an integer to each primary decoration
in **M** besides the period. In Figure 8 we have displayed the
filled Julia set
for **c = -0.12 + 0.75i**. This filled Julia set is often called Douady's
rabbit. Note that the image looks like a "fractal rabbit." The
rabbit has a main body with two ears attached. But everywhere you look
you see other pairs of ears.

** Figure 8. The fractal rabbit**

** Figure 9. A magnification of the fractal rabbit**

The fact that each junction point in this filled Julia set has 3 pieces
attached is no surprise, since this **c**-value lies in a primary period
3 bulb in the Mandelbrot set. This is another fascinating fact about
**M**. If you choose a **c**-value from one of the primary
decorations in **M**, then, first of all, **J _{c}** must be a
connected set, and second,

** Figure 10. Period 4 and 5 filled Julia sets**

Fractal Geometry of the Mandelbrot Set (Cover Page)

5 The fundamental dichotomy (Previous Section)

BU Math Home Page

Prof. Robert L. Devaney (Boston University)