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

• 
  animation
• 
  3D view of bof61
• 
  2D view of BOF61

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 than components which makes them useful for finding components)

Algorithm

whole parameter plane

// code from :
// http://mathr.co.uk/blog/2014-11-02_practical_interior_distance_rendering.html
// C program by Claude Heiland-Allen
complex double z = 0;
      complex double dc = 0;
      double minimum_z2 = infinity; // atom domain
      int period = 0;

      // iteration 
      for (int n = 1; n <= maxiters; ++n) {
        dc = 2 * z * dc + 1;
        z = z * z + c;
        double z2 = cabs2(z);

        if (z2 < minimum_z2) {
          minimum_z2 = z2;
          period = n;}}

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 :

• 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.

This algorithms shows borders of domains with the same index(c) .

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

//       function is based on function mclosetime
//       from mbrot.cpp 
//       from program mandel by Wolf Jung 
//      http:www.iram.rwth-aachen.de/~jung/indexp.html  
////8 = iterate =  bof61
// bailout2=4 
int mclosetime(std::complex<double> C , int iter_max,  int bailout2)
{ int j, cln = 0;
  double x = C.real(), y = C.imag(), u, cld = 4;
  for (j = 0; j <= iter_max; j++)
  { u = x*x + y*y; 
    if (u > bailout2) return j;
    if (u < cld) {cld = u;cln = j; }
 
    u = x*x - y*y + C.real();
    y = 2*x*y + C.imag();
    x = u;
  }
  return iter_max + cln % 15; //iterate, bof61
  
}

It can be used :

// compute escape time  
              int last_iter= mclosetime(C,iter_max,bailout2);
              //  drawing code */
              if (last_iter>=iter_max) { putpixel(ix,iy,last_iter - iter_max);} // interior
                                else putpixel(ix,iy,WHITE); // exterior

Atom domains : note that atom domains overlapp and "completely enclose the hyperbolic components of the same period"

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 :

// code by Claude Heiland-Allen
// from http://mathr.co.uk/blog/2013-12-10_atom_domain_size_estimation.html
real_t atom_domain_size_estimate(complex_t c, int_t p) {
  complex_t z = c;
  complex_t dc = 1;
  real_t abszq = cabs(z);
  for (int_t q = 2; q <= p; ++q) {
    dc = 2 * z * dc + 1;
    z = z * z + c;
    real_t abszp = cabs(z);
    if (abszp < abszq && q < p) {
      abszq = abszp;
    }
  }
  return abszq / cabs(dc);
}