Planes of M and J or F sets
Planes for Mandelbrot function Fc(z)=z*z+c (c is complex parameter, z is complex variable)
- dynamic planes = planes for Julia or Fatou sets ( c is constant; z0=z ; z is a variable, )
- parameter planes = planes for Mandelbrot Set ( z0:=0 ; c is variable )
- c-plane ( plain parameter plane )
- 1/c plane ( inverted parameter plane)
- 1/(c-0.25) plane ;
- 1/(c+1.40115) plane
- 1/(c+2.0) plane
- unrolled ( flatten) plane
map c = r/2 eip (1 - r/2 eip) with r running from 1 at the bottom to 5 at the top,
and p running from -1.1 pi to +1.1 pi left to right. The cardioid is then the straight line r=1 ( unrolled or flatten the cardioid)
- Mercator plane by David Madore :
"... it is a log map toward the target point (or, as some might say, a Mercator projection with the target point as South pole
and complex infinity as North pole); horizontally it is periodic and I have placed two periods side to side,
whereas vertically it extends to infinity at the top and at the bottom,
which corresponds to zooming infinitely far out or in, at a factor of exp(2*Pi) = 535.5
for every size of a horizontal period. Horizontal lines ( parallels ) on the log map correspond to concentric circles
around the target point, and vertical lines to radii emanating from it; and the anamorphosis preserves angles."
"Images of the Mandelbrot set with a logarithmic projection around a point c0: z-> (log(|z-c0|), arg(z-c0)).
The idea for this kind of logarithmic map of the set is from David Madore.
While it is a bit of a stretch to call it a Mercator projection, it sounds better than just a log scale map.
If the Mandelbrot set is assumed to be about meter-sized these zooms reach the size of an atomic nucleus." ( Arenamontanus / Anders Sandberg )
multiplier map of hyperbolic component converts component into unit circle
Multiplier planes for main cardioid ( H0)
- m = M(c) = 1 - sqrt(1 - 4*c)
It is a correspondence between the interior of the main cardioid and the interior of the unit disc D( disk of radius 1 centered at the origin)
- "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:
phi = 2 *p *m/n
where m/n is rotation number (e.g. 1/2 -> 180o, 1/3 -> 120o and 1/4 -> 90o) "
( quotation from: Rotation Numbers and Internal angles of the Mandelbrot bulbs by Robert L. Devaney
Planes for Flambda(z) = lambda* z*(1-z):
- The 1/(lambda-1)
Planes for Fm(z) =m*z+(1-m)*z*z
- m plane
here the (finite) fixed points of Fm(z) are:
Autor: Adam Majewski adammaj1-at-o2-dot--pl
Feel free to e-mail me. (:-))