"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 z_{o}=0 is attracted to a cycle of a period n.
For "square" parametrisation c = ^{1}/_{4} - a^{2} z_{n+1} = z_{n}^{2} + ^{1}/_{4} - a^{2} 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} → 180^{o}, ^{1}/_{3} → 120^{o} and ^{1}/_{4} → 90^{o}) |
[1] Robert L. Devaney The Fractal Geometry of the Mandelbrot Set II.
How to Count and How to Add:
3 Periods of the Bulbs