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6 Back to M

There is another way to attach an integer to each primary decoration in M besides the period. In Figure 8 we have displayed the Julia set for c = -0.12 + 0.75i. This 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

Another way to say this is that the Julia set contains infinitely many "junction points" at which 3 distinct black regions in Jc are attached. In Figure 9 we have magnified a portion of the fractal rabbit to illustrate this.

Figure 9. A magnification of the fractal rabbit

The fact that each junction point in this 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 Julia sets


 
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