Fractals/Iterations in the complex plane/atomdomains

< Fractals < Iterations in the complex plane

Name

period domain
atom domain^[1]^[2].
BOF61^[3]
orbit trap at (0,0) ^[4]

Images

animation
3D view of bof61
2D view of BOF61

Video:

2D Mandelbrot zoom (4K, 60fps) ayoungblood

Description

Atom domains in case of the Mandelbrot set ( parameter plane) are parts of parameter plane for which the index p≥1 at which |zp| is minimized during iteration of z0=0 and zn+1=z2n+c.

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"^[5]
atom domain contain :
- component of mandelbrot set with period n mv
- exterior of this component
- some other componnets

Importance

It can be used for :

fast finding of period n centers, components of Mandelbrot set

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;}}

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)^[6] ^[7].

Fragment of code : fractint.cfrm from Gnofract4d ^[8]

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"^[9]

Note that this method can be applied to both exterior and interior. It is called atom domain ^[10]. It can also be modified ^[11]

Size of the atom domain

estimation of size ^[12]

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);
}

References

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