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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, Jc must be a connected set, and second, Jc contains infinitely many special junction points and each of these points has exactly n regions attached to it, where n is exactly the period of the bulb. Figure 10 illustrates this for periods 4 and 5.
Figure 10. Period 4 and 5 filled Julia sets
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