Fractals/Iterations in the complex plane/atomdomains
< Fractals
Name
editExamples
editImages
edit-
animation
-
3D view of bof61
-
2D view of BOF61
Video
editDescription
editAtom domains in case of the Mandelbrot set ( parameter plane) are parts of parameter plane with the same the index p.
index
edit- it is positive integer
- 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
editNote that :
- atom domains are overlapping
- "Atom domains surround hyperbolic components of the same period, and are generally much larger than the components themselves"[6]
- "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"[7]
Atom domain contain :
- component of mandelbrot set with period n mv
- exterior of this component
- some other components
Importance
editIt 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
editwhole parameter plane
edit// 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
modifications
edit- Filtered atom domains
- modified atom domains - which makes smaller domains more visible.
bof61
editThis 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)[8] .[9]
Fragment of code : fractint.cfrm from Gnofract4d [10]
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
Note that this method can be applied to both exterior and interior. It is called atom domain.[12] It can also be modified [13]
Size of the atom domain
editestimation of size [14]
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
edit- ↑ Atom Domain by Robert P. Munafo
- ↑ Practical interior distance rendering by Claude Heiland-Allen
- ↑ The_Beauty_of_Fractals in english wikipedia
- ↑ Bof61 algorithm in wikibooks
- ↑ An orbit trap at (0,0) by hobold
- ↑ misiurewicz_domains by Claude
- ↑ mandelbrot set newton basins by Claude Heiland-Allen
- ↑ Fractint doc by Noel Giffin
- ↑ A Series of spiral bays in the Mandelbrot set by Patrick Hahn
- ↑ gnofract4d
- ↑ Practical interior distance rendering by Claude Heiland-Allen
- ↑ Atom Domain From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo
- ↑ Modified Atom Domains by Claude Heiland-Allen
- ↑ Atom domain size estimation by Claude Heiland-Allen