Rotation Numbers and Internal angles of
the Mandelbrot bulbs
Robert L. Devaney
The Mandelbrot set consists of many small decorations or bulbs
(or limbs or atoms) .
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 z0=0 is attracted to a cycle of a period n.
It is a multiple of n for c inside the other smaller
bulbs attached to the primary bulb.
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
f = 2
where m/n is rotation number
(e.g. 1/2 -> 180o,
1/3 -> 120o and
1/4 -> 90o)
"The Mandelbrot cactus" ("square" parametrisation).
1. One can count rotation number of a bulb by its periodic orbit star.
An attracting period n cycle z1 -> z2
->...-> zn -> z1 hops among zi as
fc is iterated. If we observe this motion, the cycle jumps
exactly m points in the counterclockwise direction at each
iteration. Another way to say this is the cycle rotates by a m/
n revolution in the counterclockwise direction under iteration.
2. The Jc-set contains infinitely many "junction points" at
which n distinct black regions in J-set are attached, because c-
value lies in a primary period n (3 or 5 for these images) bulb in the
M-set. And the smallest black region is located m revolutions in
the counterclockwise direction from the largest central region.
3. The number of spokes in the largest antenna attached to a primary
decoration is equivalent to the period of that decoration. And
the shortest spoke is located m revolutions in the counterclockwise
direction from the main spoke ("C" parametrisation here).
 Robert L. Devaney The Fractal Geometry of the Mandelbrot Set II.
How to Count and How to Add:
3 Periods of the Bulbs
Next: The primary Bulbs counting
updated 12 February 2000