Fractals/Iterations in the complex plane/atomdomains

=Name=
 * period domain
 * atom domain.
 * BOF61( see book Beauty Of Fractals page 61 )
 * orbit trap at (0,0)
 * mclosetime

=Examples=

Images

 * fractalforums : cellular-coloring-of-mandelbrot-insides

Video

 * 2D Mandelbrot zoom (4K, 60fps) ayoungblood
 * You tube video : Mandelbrot, 1080p 25fps by Warren Garabrandt

=Description= Atom domains in case of the Mandelbrot set ( parameter plane) are parts of parameter plane with the same the index p.

index

 * it is positive integer $$ p\ge 1$$
 * for p=1 domain is a whole plane because in the algorithm value of complex modulus is compared to infinity
 * it is equal to
 * the period of hyperbolic component of the Mandelbrot set which is inside domain
 * iteration at which modulus of z is minimized during iteration of critical point

=Properities= Note that :
 * atom domains are overlapping
 * "Atom domains surround hyperbolic components of the same period, and are generally much larger than the components themselves"
 * "These domains completely enclose the hyperbolic components of the same period"
 * "the atom domain is wholy within its own Newton basin, and also significantly larger than the corresponding component"

Atom domain contain :
 * component of mandelbrot set with period n mv
 * exterior of this component
 * some other components

=Importance= It can be used for :
 * fast finding ( aprioximating) of period n components of Mandelbrot set and it's centers, ( domains are greater then components which makes them useful for finding components)

=Algorithm=

whole parameter plane
Shadertoy
 * Shadertoy bof61

modifications

 * Filtered atom domains
 * modified atom domains - which makes smaller domains more visible.

bof61
This is the method described in the book "The Beauty of Fractals" on page 63, but the image in on page 61.

Color of point is proportional to : This algorithms shows borders of domains with the same index(c) .
 * the time it takes z to reach its smallest value
 * iterate of the critical point makes the closest approach
 * Index (c) is the iteration when point of the orbit was closest to the origin. Since there may be more than one, index(c) is the least such.

Fragment of code : fractint.cfrm from Gnofract4d

bof61 { init: int current_index = -1 ; -1 to match Fractint's notion of iter count int index_of_closest_point = -1 float mag_of_closest_point = 1e100 loop: current_index = current_index + 1 float zmag = |z| if zmag < mag_of_closest_point index_of_closest_point = current_index mag_of_closest_point = zmag endif final: #index = index_of_closest_point /256.0 }

Cpp function

It can be used :

Note that this method can be applied to both exterior and interior. It is called atom domain. It can also be modified

=Size of the atom domain=

estimation of size

Function for computing size estimation of atom domain from nucleus c and its period p :

=References=