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"The Mandelbrot cactus" ("square" parametrisation). |
The Mandelbrot set consists of many small decorations or bulbs
(or limbs or atoms) [1].
A decoration directly attached to the main cardioid in M is called
a primary bulb. This bulb in turn has infinitely many smaller bulbs
attached. It is known that if c lies in the interior of a bulb, then the
orbit of zo=0 is attracted to a cycle of a period n.
For "square" parametrisation c = 1/4 - a2 zn+1 = zn2 + 1/4 - a2 the main cardioid of the M-set turns into a circle with radius r = 1/2. A primary bulb attaches to the main circle at an internal angle φ = 2 π m/n where m/n is rotation number (e.g. 1/2 → 180o, 1/3 → 120o and 1/4 → 90o) |
[1] Robert L. Devaney The Fractal Geometry of the Mandelbrot Set II.
How to Count and How to Add:
3 Periods of the Bulbs