Fractals/Iterations in the complex plane/atomdomains

Name period domain atom domain.Atom Domain by Robert P. MunafoPractical interior distance rendering by Claude Heiland-Allen BOF61( see book Beauty Of Fractals page 61 )The_Beauty_of_Fractals in english wikipediaBof61 algorithm in wikibooks orbit trap at (0,0) An orbit trap at (0,0) by hobold 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 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"misiurewicz_domains by Claude "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"mandelbrot set newton basins by Claude Heiland-Allen 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)Fractint doc by Noel Giffin .A Series of spiral bays in the Mandelbrot set by Patrick Hahn Fragment of code : fractint.cfrm from Gnofract4d 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 thumbnail|Atom domains : note that atom domains overlapp and "completely enclose the hyperbolic components of the same period"Practical interior distance rendering by Claude Heiland-Allen Note that this method can be applied to both exterior and interior. It is called atom domain.Atom Domain From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo It can also be modified Modified Atom Domains by Claude Heiland-Allen Size of the atom domain estimation of size Atom domain size estimation by Claude Heiland-Allen 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