Mini Mandelbrot sets

Name:

• midget
• mini mandelbrot set
• baby Mandelbrots
• island = island mu-molecule = island mu-unit ^[1]
• primitive component

Parameter rays of mini Mandelbrot sets ^[2]

" a saddle-node (parabolic) periodic point in a complex dynamical system often admits homoclinic points, and in the case that these homoclinic points are nondegenerate, this is accompanied by the existence of infinitely many baby Mandelbrot sets converging to the saddle-node parameter value in the corresponding parameter plane." Devaney ^[3]

" Warped midgets in the Mandelbrot set have been measured, using an algorithm that allows the positions of the head, and cardioid atoms (north and south) of any midget to be found, once one has placed the cursor on the computer terminal somewhere inside any midget. We describe two distortions of midgets: linear distortions and angular distortions. When the north and south angles are plotted in the north/south angle plane, families of points are formed. The angle and distance measures of warped midgets from the Sea Horse Valley of the Mandelbrot set and from other sea horse valleys of midgets, whether on the Spike or on tendrils above atoms, all fall closely together in one part of the north/south plane. Measures of warped midgets from tendrils above the major atoms on the surface of the Cardioid fall closely together in another part of the north/south plane. This different way of looking at the Mandelbrot set offers an interesting way of studying the distortions of midgets."  A. G. Davis Philip, Michael Frame, Adam Robucci: Warped midgets in the Mandelbrot set. Computers & Graphics 18(2): 239-248 (1994)

Distortion:

• Baby Mandelbrot Sets are Not Exact Self Similar. A baby Mandelbrot sets are homeomorphic to the original one
• fractalforums : how-distorted-can-a-minibrot-be/
• math.stackexchange question: mini-mandelbrots-are-they-exact-copies ?
• math.stackexchange question: why-does-the-mandelbrot-set-contain-slightly-deformed-copies-of-itself

Julia island = Embedded Julia Set = virtual Julia Sets

• "A structure comprised of filaments, resembling a Julia set in appearance, which has a higher delta Hausdorff dimension than filaments in the immediately surrounding region. They are sometimes also called Julia islands or virtual Julia Sets." Robert Munafo^[4]
• "This location is called Julia island since it looks like a Julia set but it's actually inside Mandelbrot." ^[5]

Topological models of Mandelbrot set

Topological model of Mandelbrot set = Mandelbrot cactus

Arcs up to period 12

• Thurston model ( abstract Mandelbrot set )^[8]
  • Lavaurs algorithm ^[9]
  • Abstract Mandelbrot tree ^[10]
• Cactus model ^[11]( go to the image, where the src code and description of algorithm is )
• Fake Mandelbrot Set = M-set without hairs, filaments and primitive hyberbolic components^[12]

Symmetry

The Mandelbrot set is symmetric with respect to the x-axis in the plane :

colour(x,y) = colour(x,-y)

its intersection with the x-axis ( real slice of Mandelbrot set ) is an interval :

<-2 ; 1/4>

It can be used to speed up computations ( up to 2-times )^[13]

Bailout test

Instead of checking if magnitude ( radius = abs(z) ) is greater then escape radius ( ER):

${\displaystyle {\sqrt {Z_{x}^{2}+Z_{y}^{2}}}<ER}$

compute square of ER:

 ER2 = ER*ER

once and check :

${\displaystyle Z_{x}^{2}+Z_{y}^{2}<ER^{2}}$

It gives the same result and is faster.

interior detection

Mandelbrot set with Interior detection method

Interior detection method^[14]

• time with detection versus without detection is : 0.383s versus 8.692s so it is 23 times faster !!!!

// gives last iterate = escape time
// output 0< i < iMax
 int iterate(double complex C , int iMax)
  {
   int i=0;
   double complex Z= C; // initial value for iteration Z0
   complex double D = 1.0; // derivative with respect to z 
   
   for(i=0;i<iMax;i++)
    { if(cabs(Z)>EscapeRadius) break; // exterior
      if(cabs(D)<eps) break; // interior
      D = 2.0*D*Z;
      Z=Z*Z+C; // complex quadratic polynomial
      
    }
   return i; 
 }

Period detection

"When rendering a Mandelbrot or Julia set, the most time-consuming parts of the image are the “black areas”. In these areas, the iterations never escape to infinity, so every pixel must be iterated to the maximum limit. Therefore, much time can be saved by using an algorithm to detect these areas in advance, so that they can be immediately coloured black, rather than rendering them in the normal way, pixel by pixel. FractalNet uses a recursive algorithm to split the image up into blocks, and tests each block to see whether it lies inside a “black area”. In this way, large areas of the image can be quickly coloured black, often saving a lot of rendering time. ... (some) blocks were detected as “black areas” and coloured black immediately, without having to be rendered. (Other) blocks were rendered in the normal way, pixel by pixel." Michael Hogg ^[15]

 // cpp code by Geek3 
// http://commons.wikimedia.org/wiki/File:Mandelbrot_set_rainbow_colors.png
bool outcircle(double center_x, double center_y, double r, double x, double y)
{ // checks if (x,y) is outside the circle around (center_x,center_y) with radius r
        x -= center_x;
        y -= center_y;
        if (x * x + y * y > r * r)
                return(true);
        return(false);

 // skip values we know they are inside
                        if ((outcircle(-0.11, 0.0, 0.63, x0, y0) || x0 > 0.1)
                                && outcircle(-1.0, 0.0, 0.25, x0, y0)
                                && outcircle(-0.125, 0.744, 0.092, x0, y0)
                                && outcircle(-1.308, 0.0, 0.058, x0, y0)
                                && outcircle(0.0, 0.25, 0.35, x0, y0))
                        {
                          // code for iteration
                         }

// http://www.fractalforums.com/new-theories-and-research/quick-(non-iterative)-rejection-filter-for-mandelbrotbuddhabrot-with-benchmark/
         public static void quickRejectionFilter(BlockingCollection<Complex> input, BlockingCollection<Complex> output)
         {
            foreach(var item in input.GetConsumingEnumerable())
            {
                if ((Complex.Abs(1.0 - Complex.Sqrt(Complex.One - (4 * item))) < 1.0)) continue;
                if (((Complex.Abs(item - new Complex(-1, 0))) < 0.25)) continue;
                if ((((item.Real + 1.309) * (item.Real + 1.309)) + item.Imaginary * item.Imaginary) < 0.00345) continue;
                if ((((item.Real + 0.125) * (item.Real + 0.125)) + (item.Imaginary - 0.744) * (item.Imaginary - 0.744)) < 0.0088) continue;
                if ((((item.Real + 0.125) * (item.Real + 0.125)) + (item.Imaginary + 0.744) * (item.Imaginary + 0.744)) < 0.0088) continue;

                //We tried every known quick filter and didn't reject the item, adding it to next queue.
                output.Add(item);
            }
         }

Perturbation theory

"The thing we call perturbation consist of 2 things: Calculate one pixel with high precision and use it as a reference for all other pixels.

This method will fail though, however thanks to Pauldelbrot we have a trustable method of detecting the pixels where the reference fails to compute the pixel with hardware precision. These pixels can be rendered with a reference closer to these pixels, so a typical perturbation render use several references. This method gives a speedup at about 10 times on depths of 10^100 Use a truncated Taylor series to approximate a starting value for a specific iteration, which make you able to skip all previous iterations. This method gives a speedup of typically Another 10 times on depths of 10^100, and together the speed up is typically 100 times. This, which we call Series Approximation, is where we have issues since we do not have any solid theoretically way of finding when too many iterations are skipped - for all pixels in the view. The more terms you include in the Taylor series, the more iterations you are able to skip. So if you stay below say 50 terms, it is not likely that you ever encounter any issues. Because some views can be correctly rendered 1000 or 100,000 times faster than full precision for each pixel with many terms - can you imagine a month turned into seconds! shocked K.I.Martin originally used only 3 terms " - Kalles Fraktaler ^[17]