Question: How are the rotation numbers associated to the primary bulbs arranged around the boundary of the main cardioid? Answer: Amazingly, the order of the rotation numbers around the boundary of the main cardioid is exactly the same as the rational numbers. That is, if you call the "cusp" of the cardioid 0, and then proceed in the counterclockwise direction, you visit each bulb corresponding to rotation number p/q in exact order, ending at 1 when you arrive back at the cusp. Equally amazingly, all you need to prove this is a little (complex) calculus!

Example: To see a (QuickTime) animation of the filled Julia sets as c winds around the main cardioid (more precisely, just outside the main cardioid, sometimes in, sometimes just outside the primary bulbs), click here. Warning: This movie takes a LONG time to load---38 Mb!!! Have a lot of patience.

Exploration: Use the Mandelbrot Set Iterator together with the order of the bulbs around the main cardioid to find the following bulbs associated to the following fractions: (note that you will have to increase the number of iterations performed to around 200-300 to view magnifications of these bulbs)

• 7/19
• 14/25
• 18/31
• 25/36
• p/q where q is as large as you can actually see that this is the p/q bulb

For Further Information: See the papers:

• The Fractal Geometry of the Mandelbrot Set. II: How to Add and How to Count. Robert L. Devaney. Accessible to high school students with a solid background in complex numbers.

• The Mandelbrot and Julia Sets: A Toolkit of Dynamics Activities. Robert L. Devaney. Pages 105-114. Key Curriculum Press. Accessible to advanced high school students.

• The Mandelbrot Set and the Farey Tree. Robert L. Devaney. Postscript File. Amer. Math. Monthly 106 (1999), 289-302. Accessible to advanced math majors in college.

• Geometry of the Antennas in the Mandelbrot Set. Postscript File. Robert L. Devaney and Monica Moreno Rocha. In the journal Fractals. 10 (2002), 39-46. For graduate students in Mathematics.

Devaney home page

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