Fractals/Iterations in the complex plane/Mandelbrot set

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This book shows how to code different algorithms for drawing parameter plane [1] ( Mandelbrot set [2] ) for complex quadratic polynomial [3].

One can find different types of points / sets on parameter plane [4]


Contents

Exterior of Mandelbrot set[edit]

Colouring of exterior of Mandelbrot set can be :

  • non-smooth :
    • boolean ( bETM/M )
    • discrete ( LSM/M = iETM/M)
  • Smooth  :
    • Real Escape Time ( rETM/M )
    • Distance estimation ( DEM/M )
    • Complex potential ( CPM/M )
    • "triangle inequality"[5]
    • Orbit trap [6]

One can also draw curves :

  • external rays
  • equipotential lines ( closed curves - quasi circles)



Escape time[edit]

Here for given point c on parameter plane one checks how critical point  z = 0.0 behaves on dynamical plane under forward iteration. If you change initial point ypu will get different result [7]

To draw given plane one needs to check/scan (all) its points. See here for more details ( optimisation) Read definitions first.

Boolean escape time[edit]

Here complex plane consists of 2 sets : Mandelbrot set M\, and its complement M^c\, :

\mathbb{C}= M \cup M^c

ASCI graphic ( on screen)[edit]

ASCI graphic : Boolean escape time in text mode


-- Haskell code by Ochronus
-- http://blog.mostof.it/mandelbrot-set-in-ruby-and-haskell/
import Data.Complex
mandelbrot a = iterate (\z -> z^2 + a) a !! 500
main = mapM_ putStrLn [[if magnitude (mandelbrot (x :+ y)) < 2 then '*' else ' '
                           | x <- [-2, -1.9685 .. 0.5]]
                           | y <- [1, 0.95 .. -1]]


; common lisp
(loop for y from -1.5 to 1.5 by 0.05 do
      (loop for x from -2.5 to 0.5 by 0.025 do
		(let* ((c (complex x y)) ; parameter
                   	(z (complex 0 0))
                   	(iMax 20) ; maximal number of iterations
			(i 0)) ; iteration number
 
		(loop  	while (< i iMax ) do 
			(setq z (+ (* z z) c)) ; iteration
			(incf i)
			(when (> (abs z) 2) (return i)))
	   ; color of pixel
           (if (= i iMax) (princ (code-char 42)) ; inside M
                          (princ (code-char 32))))) ; outside M
      (format t "~%")) ; new line

Comparison programs in various languages [8]

Graphic file ( PPM )[edit]

Here are various programs for creating pbm file [9]

C[edit]

This is complete code of C one file program.

  • It makes a ppm file which consists an image. To see the file (image) use external application ( graphic viewer).
  • Program consists of 3 loops:
    • iY and iX, which are used to scan rectangle area of parameter plane
    • iterations.

For each point of screen (iX,iY) it's complex value is computed c=cx+cy*i.

For each point c is computed iterations of critical point z_0 = z_{cr} = 0 \,

It uses some speed_improvement. Instead of checking :

sqrt(Zx2+Zy2)<ER

it checks :

(Zx2+Zy2)<ER2 // ER2 = ER*ER

It gives the same result but is faster.



 /* 
 c program:
 --------------------------------
  1. draws Mandelbrot set for Fc(z)=z*z +c
  using Mandelbrot algorithm ( boolean escape time )
 -------------------------------         
 2. technique of creating ppm file is  based on the code of Claudio Rocchini
 http://en.wikipedia.org/wiki/Image:Color_complex_plot.jpg
 create 24 bit color graphic file ,  portable pixmap file = PPM 
 see http://en.wikipedia.org/wiki/Portable_pixmap
 to see the file use external application ( graphic viewer)
  */
 #include <stdio.h>
 #include <math.h>
 int main()
 {
          /* screen ( integer) coordinate */
        int iX,iY;
        const int iXmax = 800; 
        const int iYmax = 800;
        /* world ( double) coordinate = parameter plane*/
        double Cx,Cy;
        const double CxMin=-2.5;
        const double CxMax=1.5;
        const double CyMin=-2.0;
        const double CyMax=2.0;
        /* */
        double PixelWidth=(CxMax-CxMin)/iXmax;
        double PixelHeight=(CyMax-CyMin)/iYmax;
        /* color component ( R or G or B) is coded from 0 to 255 */
        /* it is 24 bit color RGB file */
        const int MaxColorComponentValue=255; 
        FILE * fp;
        char *filename="new1.ppm";
        char *comment="# ";/* comment should start with # */
        static unsigned char color[3];
        /* Z=Zx+Zy*i  ;   Z0 = 0 */
        double Zx, Zy;
        double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
        /*  */
        int Iteration;
        const int IterationMax=200;
        /* bail-out value , radius of circle ;  */
        const double EscapeRadius=2;
        double ER2=EscapeRadius*EscapeRadius;
        /*create new file,give it a name and open it in binary mode  */
        fp= fopen(filename,"wb"); /* b -  binary mode */
        /*write ASCII header to the file*/
        fprintf(fp,"P6\n %s\n %d\n %d\n %d\n",comment,iXmax,iYmax,MaxColorComponentValue);
        /* compute and write image data bytes to the file*/
        for(iY=0;iY<iYmax;iY++)
        {
             Cy=CyMin + iY*PixelHeight;
             if (fabs(Cy)< PixelHeight/2) Cy=0.0; /* Main antenna */
             for(iX=0;iX<iXmax;iX++)
             {         
                        Cx=CxMin + iX*PixelWidth;
                        /* initial value of orbit = critical point Z= 0 */
                        Zx=0.0;
                        Zy=0.0;
                        Zx2=Zx*Zx;
                        Zy2=Zy*Zy;
                        /* */
                        for (Iteration=0;Iteration<IterationMax && ((Zx2+Zy2)<ER2);Iteration++)
                        {
                            Zy=2*Zx*Zy + Cy;
                            Zx=Zx2-Zy2 +Cx;
                            Zx2=Zx*Zx;
                            Zy2=Zy*Zy;
                        };
                        /* compute  pixel color (24 bit = 3 bytes) */
                        if (Iteration==IterationMax)
                        { /*  interior of Mandelbrot set = black */
                           color[0]=0;
                           color[1]=0;
                           color[2]=0;                           
                        }
                     else 
                        { /* exterior of Mandelbrot set = white */
                             color[0]=255; /* Red*/
                             color[1]=255;  /* Green */ 
                             color[2]=255;/* Blue */
                        };
                        /*write color to the file*/
                        fwrite(color,1,3,fp);
                }
        }
        fclose(fp);
        return 0;
 }

Integer escape time = LSM/M[edit]

Here color is proportional to last iteration.

This is also called Level Set Method ( LSM )

L_n= \{ c : z_n \in T ~~\mbox{and} ~~ z_k \notin T ~~\mbox{where}~~ k<n \}\,


C[edit]

LSM/M image with full code in C

Difference between Mandelbrot algorithm and LSM/M is in only in part instruction, which computes pixel color of exterior of Mandelbrot set. In LSM/M is :

 if (Iteration==IterationMax)
   { /* interior of Mandelbrot set = black */
     color[0]=0;
     color[1]=0;
     color[2]=0;                           
   }
   /* exterior of Mandelbrot set = LSM */
   else if ((Iteration%2)==0) 
             { /* even number = black */
             color[0]=0; /* Red */
             color[1]=0; /* Green */ 
             color[2]=0; /* Blue */
             }
           else 
             {/* odd number =  white */
             color[0]=255; /* Red */
             color[1]=255; /* Green */ 
             color[2]=255; /* Blue */    
             };


Here is faster C function which can be used instead of above C++ function:

int GiveEscapeTime(double C_x, double C_y, int iMax, double _ER2)
{ 
    int i;
    double Zx, Zy;
    double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
 
    Zx=0.0; /* initial value of orbit = critical point Z= 0 */
    Zy=0.0;
    Zx2=Zx*Zx;
    Zy2=Zy*Zy;
 
    for (i=0;i<iMax && ((Zx2+Zy2)<_ER2);i++)
    {
      Zy=2*Zx*Zy + C_y;
      Zx=Zx2-Zy2 +C_x;
      Zx2=Zx*Zx;
      Zy2=Zy*Zy;
    };
 return i;
}

C++[edit]

Here is C++ function which can be used to draw LSM/M :

 int iterate_mandel(complex C , int imax, int bailout)
  {
   int i;
   std::complex Z(0,0); // initial value for iteration Z0
 
   for(i=0;i<=imax-1;i++)
    {
      Z=Z*Z+C; // overloading of operators
      if(abs(Z)>bailout)break;
    }
   return i;
 }

I think that it can't be coded simpler (it looks better than pseudocode), but it can be coded in other way which can be executed faster .

Here is faster code :

// based on cpp code by Geek3
inline int fractal(double cx, double cy, int max_iters)
// gives last iteration 
{
	double zx = 0, zy = 0;	
	if (zx * zx + zy * zy > 4) return(0); // it=0
	for (int it = 1; it < max_iters; it++)
	{	double zx_old = zx;
		zx = zx * zx - zy * zy;
		zy = 2 * zx_old * zy;
		zx += cx;
		zy += cy;
		if (zx * zx + zy * zy > 4.0) return(it);
	}
	return(max_iters);
}

Java[edit]

//Java code by Josef Jelinek 
// http://java.rubikscube.info/
 
 int mandel(double cx, double cy) {
    double zx = 0.0, zy = 0.0;
    double zx2 = 0.0, zy2 = 0.0;
    int iter = 0;
    while (iter < iterMax && zx2 + zy2 < 4.0) {
      zy = 2.0 * zx * zy + cy;
      zx = zx2 - zy2 + cx;
      zx2 = zx * zx;
      zy2 = zy * zy;
      iter++;
    }
    return iter;
  }

Java Script[edit]

Here is JavaScript function which does not give last iteration but LastIteration modulo maxCol. It makes colour cycling ( if maxCol < maxIt ).

function iterate(Cr,Ci) {
// JavaScript function by Evgeny Demidov
// http://www.ibiblio.org/e-notes/html5/fractals/mandelbrot.htm
  var I=0, R=0,  I2=0, R2=0,   n=0;
  if (R2+I2 > max) return 0;
  do  {  I=(R+R)*I+Ci;  R=R2-I2+Cr;  R2=R*R;  I2=I*I;  n++;
  } while ((R2+I2 < max) && (n < maxIt) );
  if (n == maxIt) return maxCol;  else return n % maxCol;
}

Above functions do not use explicit definition of complex number.

Lisp program[edit]

Whole Lisp program making ASCI graphic based on code by Frank Buss [10] [11]


; common lisp
(loop for y from -1.5 to 1.5 by 0.1 do
      (loop for x from -2.5 to 0.5 by 0.04 do
            (let* ((i 0)
                   (z (complex x y))
                   (c z))
              (loop while (< (abs
                              (setq z (+ (* z z) c)))
                             2)
                    while (< (incf i) 32))
              (princ (code-char (+ i 32))))) ; ASCII chars <= 32 contains non-printing characters
      (format t "~%"))


MathMap plugin for Gimp[edit]

filter mandelbrot (gradient coloration)
    c=ri:(xy/xy:[X,X]*1.5-xy:[0.5,0]);
    z=ri:[0,0]; # initial value z0 = 0 
    # iteration of z
    iter=0;
    while abs(z)<2 && iter<31
    do
        z=z*z+c;  # z(n+1) = fc(zn)
        iter=iter+1
    end;
    coloration(iter/32) # color of pixel
end

Pov-Ray[edit]

Pov-Ray has a built-in function mandel[12]

Matemathica[edit]

Here is code by Paul Nylander

Level Curves of escape time Method = LCM/M[edit]

edge detection of Level sets
Lemniscates of Mandelbrot set

Lemniscates are boundaries of Level Sets of escape time ( LSM/M ). They can be drawn using 2 methods:

  • edge detection of Level sets.
    • Algorithm described in paper by M. Romera et al.[13] This method is fast and allows looking for high iterations.
  • boundary trace[14]
  • drawing curves L_n(T)=\{c: abs(z_n)=ER  \}\,, see explanation and source code. This method is very complicated for iterations > 5.

Decomposition of exterior of Mandelbrot set[edit]

Decomposition is modification of escape time algorithm.

The target set is divided into parts (2 or more). Very large escape radius is used, for example ER = 12.

Binary decomposition of LSM/M[edit]

binary decomposition: image with full code in C

Here target set T\, on dynamic plane is divided into 2 parts (binary decomposition = 2-decomposition ):

  • upper half ( white) T_u = \{ z : |z| > ER ~~\mbox{and}~~ Im (z) > 0\}\,
  • lower half (black) T_l = \{ z : |z| > ER ~~\mbox{and}~~ Im (z) \le 0 \}\,

Division of target set induces decomposition of level sets L_n\, into 2^{n+1}\, parts:

  • L_{n,u}  =\{ c:  |z_n| > ER ~~\mbox{and}~~ Im (z_n) > 0 \}\, which is colored white,
  • L_{n,l} = \{ c : |z_n| > ER ~~\mbox{and}~~ Im (z_n) \le 0 \}\, which is colored black.


External rays of angles (measured in turns):

angle = (k / 2^n ) ~~\mbox{mod }~1\,

can be seen.

Difference between binary decomposition algorithm and Mandel or LSM/M is in only in part of instruction , which computes pixel color of exterior of Mandelbrot set. In binary decomposition is :

 if (Iteration==IterationMax)
  { /* interior of Mandelbrot set = black */
   color[0]=0;
   color[1]=0;
   color[2]=0;           
  }
  /* exterior of Mandelbrot set = LSM */
  else if (Zy>0) 
         { 
          color[0]=0; /* Red */
          color[1]=0; /* Green */ 
          color[2]=0; /* Blue */
         }
        else 
          {
           color[0]=255; /* Red */
           color[1]=255; /* Green */ 
           color[2]=255; /* Blue */    
          };

Point c is plotting white or black if imaginary value of last iteration ( Zy) is positive or negative.[15]

nth-decomposition[edit]

This method is extension of binary decomposition.

The target set T = { z : |zn| > R } with a very large escape radius ( for example R = 12 ) is divided into more then 2 parts ( for example 8).[16]

Real Escape Time[edit]

Other names of this method/algorithm are :

  • Normalized Iteration Count Algorithm
  • Continuous coloring
  • smooth colour gradient
  • fractional iterations

Here color of exterior of Mandelbrot set is proportional not to Last Iteration ( which is integer number) but to real number :

\nu(z) = \lim_{i \to \infty} (i - \log_2 \log_2 |z_i|)\,

Other methods and speedups

Colouring formula in Ultrafractal :[17]

smooth iter = iter + 1 + ( log(log(bailout)-log(log(cabs(z))) )/log(2)

where :

  • log(log(bailout) can be precaclulated

C[edit]

To use log2 function add :

#include <math.h>

at the beginning of program.


if (Iteration==IterationMax)
 { /*  interior of Mandelbrot set = black */
  color[0]=0;
  color[1]=0;
  color[2]=0;                           
 }
 /* exterior of Mandelbrot set  */
 else GiveRainbowColor((double)(Iteration- log2(log2(sqrt(Zx2+Zy2))))/IterationMax,color);

where :

  • Zx2 = Zx*Zx
  • Zy2 = Zy*Zy


Here is another version by Tony Finch[18]

while (n++ < max &&
 x2+y2 < inf) {
y = 2*x*y + b;
x = x2-y2 + a;
y2 = y*y;
x2 = x*x;
}
nu = n - log(log(x2+y2)/2)/ log(2);

based on equation [19]

\nu(z) = n - \log_2 \log (z_n)\,

C++[edit]

// based on cpp code by Geek3 from http://en.wikibooks.org/wiki/File:Mandelbrot_set_rainbow_colors.png
sqrxy = x * x + y * y;
double m = LastIteration + 1.5 - log2(log2(sqrxy));

Matemathica[edit]

Here is code by Paul Nylander. It uses different formula :

cet = n + log_2ln(R) - log_2ln|z|

Python[edit]

Python code using mpmath library[20]

def mandelbrot(z):
    c = z
    for i in xrange(ITERATIONS):
        zprev = z
        z = z*z + c
        if abs(z) > ESCAPE_RADIUS:
            return ctx.exp(1j*(i + 1 - ctx.log(ctx.log(abs(z)))/ctx.log(2)))
    return 0

Distance estimation DEM/M[edit]

Variants :

  • exterior DEM/M
  • interior DEM/M

Description

Complex potential[edit]

Uniformization of complement of Mandelbrot set

Complex potential is a complex number, so it has a real (real potential) and an imaginary part(external angle). One can take its curl, divergence or its absolute value. So on one image one can use more than one variable to color image.[21]

Real potential = CPM/M[edit]

In Fractint :

potential =  log(modulus)/2^iterations

One can use real potential to:

  • smooth (continuous) coloring[22]
  • discrete coloring ( level sets of potential)
  • 3D view

Here is Delphi function which gives level of potential :


 Function GiveLevelOfPotential(potential:extended):integer;
  var r:extended;
  begin
     r:= log2(abs(potential));
     result:=ceil(r);
  end;

External angle and external ( parameter) ray[edit]

First find angle of last iteration. It is easy to compute and shows some external rays as a borders of level sets.

Then one go futher :

Tests[edit]

The Wolf Jung test : The external parameter rays for angles (in turns)

  • 1/7 (period 3)
  • 321685687669320/2251799813685247 (period 51)
  • 321685687669322/2251799813685247 ( period 51 )

Angles differ by about 10^{-15}, but the landing points of the corresponding parameter rays are about 0.035 apart. [25]

The test by G. Pastor and Miguel Romera The external parameter rays for angles (in turns)

  • 6871947673/34359738367 ( period 35 )
  • 9162596898/34359738367 ( period 35 )

the central babies Mandelbrot sets of the cauliflowers located at -0.153756141 + 1.030383223i

(not that 34359738367 = 2^35 - 1)

Boundary of Mandelbrot set[edit]

Boundary of Mandelbrot set as an image of unit circle under \Psi_M\,

Mini Mandelbrot sets[edit]

" 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 [26]

Drawing boundaries[edit]

Methods used to draw boundary of Mandelbrot set :[27]

Topological models of Mandelbrot set[edit]

Topological model of Mandelbrot set = Mandelbrot cactus
Arcs up to period 12
  • Thurston model ( abstract Mandelbrot set )[32]
  • Cactus model [35]( go to the image, where the src code and description of algorithm is )
  • Fake Mandelbrot Set = M-set without hairs, filaments and primitive hyberbolic componennts[36]

Computing Misiurewicz points of complex quadratic mapping[edit]

"... we do not know how to compute (...) Misiurewicz parameters (with high (pre)periods) for the family of quadratic rational maps. One might need to and a non-rigorous method to and Misiurewicz parameter in a reasonable time like Biham-Wenzel's method." HIROYUKI INOU [37]

First method[edit]

Misiurewicz points [38] are special boundary points.

Define polynomial in Maxima CAS :

P(n):=if n=0 then 0 else P(n-1)^2+c;

Define a Maxima CAS function whose roots are Misiurewicz points, and find them.

M(preperiod,period):=allroots(%i*P(preperiod+period)-%i*P(preperiod));

Examples of use :

(%i6) M(2,1);
(%o6) [c=-2.0,c=0.0]
(%i7) M(2,2);
(%o7) [c=-1.0*%i,c=%i,c=-2.0,c=-1.0,c=0.0]

Second Method[edit]

" factorizing the polynomials that determine Misiurewicz points. I believe that you should start with

  ( f^(p+k-1) (c) + f^(k-1) (c) ) / c

This should already have exact preperiod k , but the period is any divisor of p . So it should be factorized further for the periods.

Example: For preperiod k = 1 and period p = 2 we have

  c^3 + 2c^2 + c + 2 .

This is factorized as

(c + 2)*(c^2 + 1)  

for periods 1 and 2 . I guess that these factors appear exactly once and that there are no other factors, but I do not know."[39]

Interior of Mandelbrot set - hyperbolic components[edit]

The Lyapunov Exponent[edit]

Lyapunov exponents of mini the Mandelbrot set
Lyapunov exponent of real quadratic map

Math equation :[40]

\lambda_f(z_0) = \lim_{n\rightarrow\infty}  \frac{1}{n} \sum_{i=0}^{n-1} \left ( \ln\left|f'(z_i)\right | \right )

where :

f'(x) = \frac{d}{dz}f_c(z) = 2z

means first derivative of f with respect to z


See also :

  • image and description by janthor [41]
  • image by Anders Sandberg [42]

Interior distance estimation[edit]

Interior distance estimation

Description of method

absolute value of the orbit[edit]

# Hypercomputing the Mandelbrot Set? by Petrus H. Potgieter February 1, 2008
n=1000; # For an nxn grid
m=50; # Number of iterations
c=meshgrid(linspace(-2,2,n))\ # Set up grid
+i*meshgrid(linspace(2,-2,n));
x=zeros(n,n); # Initial value on grid
for i=1:m
x=x.^2+c; # Iterate the mapping
endfor
imagesc(min(abs(x),2.1)) # Plot monochrome, absolute
# value of 2.1 is escape


internal level sets[edit]

Color of point  :

  • is proportional to the value of z is at final iteration.
  • shows internal level sets of periodic attractors.


bof60[edit]

Image of bof60 in on page 60 in the book "the Beauty Of Fractals".Description of the method described on page 63 of bof. It is used only for interior points of the Mandelbrot set.

Color of point is proportional to :

  • the smallest distance of its orbit from origin[43][44]
  • the smallest value z gets during iteration [45]
  • illuminating the closest approach the iterates of the origin (critical point) make to the origin inside the set


Level sets of distance are sets of points with the same distance[46]

if (Iteration==IterationMax)
 /* interior of Mandelbrot set = color is proportional to modulus of last iteration */
 else { /* exterior of Mandelbrot set = black */
  color[0]=0;
  color[1]=0;
  color[2]=0;                           
 }
  • fragment of code : fractint.cfrm from Gnofract4d [47]
bof60 {
 init:
       float mag_of_closest_point = 1e100
 loop:
       float zmag = |z|
       if zmag < mag_of_closest_point
               mag_of_closest_point = zmag
       endif
 final:
       #index = sqrt(mag_of_closest_point) * 75.0/256.0
}

bof61[edit]

3D view of bof61
2D view of 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)[48] [49].


Fragment of code : fractint.cfrm from Gnofract4d [50]

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

Period of hyperbolic components[edit]

period of hyperbolic components

Period of hyperbolic component of Mandelbrot set is a period of limit set of critical orbit.


Algorithms for computing period:

  • direct period detection from iterations of critical point z = 0.0 on dynamical plane
  • "quick and dirty" algorithm : check if abs(z_n ) < eps   then colour c-point with colour n. Here n is a period of attracting orbit and eps is a radius of circle around attracting point = precision of numerical computations
  • "methods based on interval arithmetic when implemented properly are capable of finding all period-n cycles for considerable large n." (ZBIGNIEW GALIAS )[51]
  • Floyd's cycle-finding algorithm [52]
  • the spider algorithm

Multiplier map[edit]

definition

Internal angle[edit]

Method by Renato Fonseca : [53] "a point c in the set is given a hue equal to argument

arg(z_{n_{max}}) = arctan\frac{Im(z_{n_{max}})}{Re(z_{n_{max}})}

(scaled appropriatly so that we end up with a number in the range 0 - 255). The number z_nmax is the last one calculated in the z's sequence. "

Internal rays[edit]

Internal and external rays

When radius\, varies and angle\, is constant then c\, goes along internal ray. It is used as a path inside Mandelbrot set

/* find c in component of Mandelbrot set 
 uses complex type so #include <complex.h> and -lm 
 uses code by Wolf Jung from program Mandel
 see function mndlbrot::bifurcate from mandelbrot.cpp
 http://www.mndynamics.com/indexp.html
 
  */
double complex GiveC(double InternalAngleInTurns, double InternalRadius, unsigned int period)
{
  //0 <= InternalRay<= 1
  //0 <= InternalAngleInTurns <=1
  double t = InternalAngleInTurns *2*M_PI; // from turns to radians
  double R2 = InternalRadius * InternalRadius;
  double Cx, Cy; /* C = Cx+Cy*i */
  switch ( period ) {
    case 1: // main cardioid
      Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; 
      Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; 
      break;
   case 2: // only one component 
      Cx = InternalRadius * 0.25*cos(t) - 1.0;
      Cy = InternalRadius * 0.25*sin(t); 
      break;
  // for each period  there are 2^(period-1) roots. 
  default: // safe values
      Cx = 0.0;
      Cy = 0.0; 
    break; }
 
  return Cx+ Cy*I;
}
 
// draws points to memmory array data
int DrawInternalRay(double InternalAngleInTurns , unsigned int period, int iMax, unsigned char data[])
{
 
   complex double c;
   double InternalRadius;
   double RadiusStep; // between radius of points 
   int i; // number of point to draw
 
  RadiusStep = 1.0/iMax;
 
  for(i=0;i<=iMax;++i){ 
   InternalRadius = i * RadiusStep;
   c = GiveC(InternalAngleInTurns, InternalRadius, period);
   DrawPoint(c,data);
  }
 
return 0;
}

Example : internal ray of angle =1/6 of main cardioid.

Internal angle :

angle = 1/6 \,

radius of ray :

 0 \le radius \le 1  \,

Point of internal radius of unit circle :

 w = radius * e^{i * angle}\,

Map point w to parameter plane :

c = \frac{w}{2} - \frac{w^2}{4} \,

For epsilon = 0 \, this is equation for main cardioid.

Internal curve[edit]

When radius\, is constant varies and angle\, varies then c\, goes along internal curve.

/* find c in component of Mandelbrot set 
 uses complex type so #include <complex.h> and -lm 
 uses code by Wolf Jung from program Mandel
 see function mndlbrot::bifurcate from mandelbrot.cpp
 http://www.mndynamics.com/indexp.html
 
  */
double complex GiveC(double InternalAngleInTurns, double InternalRadius, unsigned int period)
{
  //0 <= InternalRay<= 1
  //0 <= InternalAngleInTurns <=1
  double t = InternalAngleInTurns *2*M_PI; // from turns to radians
  double R2 = InternalRadius * InternalRadius;
  double Cx, Cy; /* C = Cx+Cy*i */
  switch ( period ) {
    case 1: // main cardioid
      Cx = (cos(t)*InternalRadius)/2-(cos(2*t)*R2)/4; 
      Cy = (sin(t)*InternalRadius)/2-(sin(2*t)*R2)/4; 
      break;
   case 2: // only one component 
      Cx = InternalRadius * 0.25*cos(t) - 1.0;
      Cy = InternalRadius * 0.25*sin(t); 
      break;
  // for each period  there are 2^(period-1) roots. 
  default: // safe values
      Cx = 0.0;
      Cy = 0.0; 
    break; }
 
  return Cx+ Cy*I;
}
 
 
// draws points to memmory array data
int DrawInternalCurve(double InternalRadius , unsigned int period,  int iMax, unsigned char data[])
{
 
   complex double c;
   double InternalAngle; // in turns = from 0.0 to 1.0
   double AngleStep;
   int i;
   // int iMax =100;
 
  AngleStep = 1.0/iMax;
 
  for(i=0;i<=iMax;++i){ 
   InternalAngle = i * AngleStep;
   c = GiveC(InternalAngle, InternalRadius, period);
   DrawPoint(c,data);
  }
 
return 0;
}

Boundaries of hyperbolic components[edit]

Boundaries of hyperbolic components for periods 1-6 as a solutions of boundary equations

Methods of drawing boundaries:

  • solving boundary equations :
    • using method by Brown,Stephenson,Jung . It works only up to period 7-8 [54]
    • using resultants [55]
  • parametrisation of boundary with Newton method near centers of components[56] [57]. This methods needs centers, so it has some limitations,
  • Boundary scanning  : detecting the edges after detecting period by checking every pixel. This method is slow but allows zooming. Draws "all" components
  • Boundary tracing for given c value. Draws single component.
  • Fake Mandelbrot set by Anne M. Burns : draws main cardioid and all its descendants. Do not draw mini Mandelbrot sets. [58]


"... to draw the boundaries of hyperbolic components using Newton's method. That is, take a point in the hyperbolic component that you are interested in (where there is an attracting cycle), and then find a curve along which the modulus of the multiplier tends to one. Then you will have found an indifferent parameter. Now you can similarly change the argument of the multiplier, again using Newton's method, and trace this curve. Some care is required near "cusps". " Lasse Rempe-Gillen[59]


solving boundary equations[edit]

System of 2 equations defining boundaries of period n\, hyperbolic components[edit]
  • first defines periodic point,
  • second defines indifferent orbit ( multiplier of periodic point is equal to one ).

\begin{cases}
    F(n,z,c) = z \\
    abs(\lambda (z) )=1 
\end{cases}

Because stability index abs(\lambda)\, is equal to radius of point of unit circle abs(w)\,:

abs(\lambda) = abs(w)\,

so one can change second equation to form [60] :

\lambda = w \,

It gives system of equations :

\begin{cases}
     F(n,z,c) = z \\
    \lambda = w 
\end{cases}


It can be used for :

  • drawing components ( boundaries, internal rays )
  • finding indifferent parameters ( parabolic or for Siegel discs )



Above system of 2 equations has 3 variables : z,c,w \, ( n\, is constant and multiplier \lambda\, is a function of z,c \,). One have to remove 1 variable to be able to solve it.

Boundaries are closed curves : cardioids or circles. One can parametrize points of boundaries with angle t\, ( here measured in turns from 0 to 1 ).

After evaluation of w = l(t) \, one can put it into above system, and get a system of 2 equations with 2 variables z,c\,.

Now it can be solved


For periods:

  • 1-3 it can be solved with symbolical methods and give implicit ( boundary equation) b_p(w,c)=0 \, and explicit function (inverse multiplier map)  : c=\gamma_p(w)\,
    • 1-2 it is easy to solve [61]
    • 3 it can be solve using "elementary algebra" ( Stephenson )
  • >3 it can't be reduced to explicitly function but
    • can be reduced to implicit solution ( boundary equation) b_p(w,c)=0 \, and solved numerically
    • can be solved numerically using Newton method for solving system of nonlinear equations


Solving system of equation for period 1[edit]

Here is Maxima code :

(%i4) p:1;
(%o4) 1
(%i5) e1:F(p,z,c)=z;
(%o5) z^2+c=z
(%i6) e2:m(p)=w;
(%o6) 2*z=w
(%i8) s:eliminate ([e1,e2], [z]);
(%o8) [w^2-2*w+4*c]
(%i12) s:solve([s[1]], [c]);
(%o12) [c=-(w^2-2*w)/4]
(%i13) define (m1(w),rhs(s[1]));
(%o13) m1(w):=-(w^2-2*w)/4


Second equation contains only one variable, one can eliminate this variable. Because boundary equation is simple so it is easy to get explicit solution

m1(w):=-(w^2-2*w)/4
Solving system of equation for period 2[edit]

Here is Maxima code using to_poly_solve package by Barton Willis:

(%i4) p:2;
(%o4) 2
(%i5) e1:F(p,z,c)=z;
(%o5) (z^2+c)^2+c=z
(%i6) e2:m(p)=w;
(%o6) 4*z*(z^2+c)=w
(%i7) e1:F(p,z,c)=z;
(%o7) (z^2+c)^2+c=z
(%i10) load(topoly_solver);
to_poly_solve([e1, e2], [z, c]);
(%o10) C:/PROGRA~1/MAXIMA~1.1/share/maxima/5.16.1/share/contrib/topoly_solver.mac
(%o11) [[z=sqrt(w)/2,c=-(w-2*sqrt(w))/4],[z=-sqrt(w)/2,c=-(w+2*sqrt(w))/4],[z=(sqrt(1-w)-1)/2,c=(w-4)/4],[z=-(sqrt(1-w)+1)/2,c=(w-4)/4]]
(%i12) s:to_poly_solve([e1, e2], [z, c]);
(%o12) [[z=sqrt(w)/2,c=-(w-2*sqrt(w))/4],[z=-sqrt(w)/2,c=-(w+2*sqrt(w))/4],[z=(sqrt(1-w)-1)/2,c=(w-4)/4],[z=-(sqrt(1-w)+1)/2,c=(w-4)/4]]
(%i14) rhs(s[4][2]);
(%o14) (w-4)/4
(%i16) define (m2 (w),rhs(s[4][2]));
(%o16) m2(w):=(w-4)/4

explicit solution :

m2(w):=(w-4)/4
Solving system of equation for period 3[edit]

For period 3 ( and higher) previous method give no results (Maxima code) :

(%i14) p:3;
e1:z=F(p,z,c);
e2:m(p)=w;
load(topoly_solver);
to_poly_solve([e1, e2], [z, c]);
(%o14) 3
(%o15) z=((z^2+c)^2+c)^2+c
(%o16) 8*z*(z^2+c)*((z^2+c)^2+c)=w
(%i17) 
(%o17) C:/PROGRA~1/MAXIMA~1.1/share/maxima/5.16.1/share/contrib/topoly_solver.mac
`algsys' cannot solve - system too complicated.
#0: to_poly_solve(e=[z = ((z^2+c)^2+c)^2+c,8*z*(z^2+c)*((z^2+c)^2+c) = w],vars=[z,c])
-- an error.  To debug this try debugmode(true);

I use code by Robert P. Munafo[62] which is based on paper of Wolf Jung.

Boundary equation[edit]

The result of solving above system with respect to c\, is boundary equation,


b_p(w,c)=0 \,

where b_p(w,c)\, is boundary polynomial.

It defines exact coordinates of hyperbolic componenets for given period p \,.

It is implicit equation.

Period 1

One can easly compute boundary point c

 c = c_x + c_y*i

of period 1 hyperbolic component ( main cardioid) for given internal angle ( rotation number) t using this code by Wolf Jung[63]

t *= (2*PI); // from turns to radians
cx = 0.5*cos(t) - 0.25*cos(2*t); 
cy = 0.5*sin(t) - 0.25*sin(2*t); 

Period 2

t *= (2*PI);  // from turns to radians
cx = 0.25*cos(t) - 1.0;
cy = 0.25*sin(t);  

Solving boundary equations[edit]

Solving boundary equations for various angles gives list of boundary points.

Centers of components[edit]

Methods of finding centers :


  • all centers for given period p\,
    • solving F_p(0,c) = 0 \, or   G_p(0,c) = 0\,
      • showing results in graphical form
      • using numerical methods[64]
        • "quick and dirty" algorithm : check if abs(z_n ) < eps then colour c-point with colour n. Here n is a period of attracting orbit and eps is a radius of circle around attracting point = precision of numerical comutations
        • finding all roots of polynomial
        • Myrberg's method [65][66]
      • interval arithmetics and a divide and conquere strategy[67]
  • one center c_p\, for given period p\, near given point c\,


System of equations defining centers[edit]

Centers of hyperbolic components for period 10
Centers of hyperbolic components for period 1-10 computed using irreducible polynomials

Center of hyperbolic component is a point c\, of parameter plain, for which periodic z-cycle is superattracting. It gives a system of 2 equations defining centers of period p\, hyperbolic components :

  • first defines periodic point z_0\,,
  • second makes z_0\, point superattracting.

\begin{cases}
    F_p(z_0,c) = z_0 \\
     \frac{df_c^{(p)}(z_0)}{dz}=0 
\end{cases}


see definitions for details.

Solving system of equations[edit]

Second equation can be solved when critical point z_{cr} = 0 \, is in this cycle :

z_0 = z_{cr} = 0 \,

To solve system put z_0\, into first equation.

Equation defining centers[edit]

One gets equation :

F_p(0,c) = 0 \,

Centers of components are roots of above equation.

Because z = 0 \, one can remove z from these equations :

  • for period 1 : z^2+c=z and z=0 so c=0
  • for period 2 : (z^2+c)^2 +c =z and z=0 so c^2 + c =0
  • for period 3 : ((z^2+c)^2 +c)^2 +c = z and z=0 so (c^2 + c)^2 +c =0

Here is Maxima functions for computing above functions :

P[n]:=if n=1 then c else P[n-1]^2+c;
Reduced equation defining centers[edit]

Set of roots of above equation contains centers for period p and its divisors. It is because :

F_p(0,c) = \prod_{m|p} G_m(0,c) \,

where :

For example :


F_1 = c = G_1 \,

F_2 = c*(c+1) = G_1* G_2 \,

F_3 = c*(c^3+2*c^2+c+1) = G_1* G_3 \,

F_4 = c*(c+1)*(c^6+3*c^5+3*c^4+3*c^3+2*c^2+1) = G_1* G_2* G_4 \,

So one can find irreducible polynomials using :


G_p(0,c) = \frac{F_p(0,c) }{ \prod_{m|p,m<p} G_m(0,c)}  \,


Here is Maxima function for computing G \, :


GiveG[p]:=
block(
[f:divisors(p),
t:1], /* t is temporary variable = product of Gn for (divisors of p) other than p */
f:delete(p,f), /* delete p from list of divisors */
if p=1
then return(Fn(p,0,c)),
for i in f do t:t*GiveG[i],
g: Fn(p,0,c)/t,
return(ratsimp(g))
)$

Here is a table with degree of F\, and G\, for periods 1-10 and precision needed to compute roots of these functions.


period p \, \deg(F_p(0,c))  \, \deg(G_p(0,c))  \, fpprec \,
1 1 1 16
2 2 1 16
3 4 3 16
4 8 6 16
5 16 15 16
6 32 27 16
7 64 63 32
8 128 120 64
9 256 252 128
10 512 495 300
11 1024 1023 600

Here is a table of greatest coefficients.

period p \, log10(gc_p)\, gc_p \,
1 0 1
2 0 1
3 1 2
4 1 3
5 3 116
6 4 5892
7 11 17999433372
8 21 106250977507865123996
9 44 16793767022807396063419059642469257036652437
10 86 86283684091087378792197424215901018542592662739248420412325877158692469321558575676264
11 179 307954157519416310480198744646044941023074672212201592336666825190665221680585174032224052483643672228833833882969097257874618885560816937172481027939807172551469349507844122611544

Precision can be estimated as bigger than size of binary form of greatest coefficient log_2(gc)\, :

fpprec > log_2(gc)\,
period p \, \deg(G_p(0,c))  \, \log_2(gc_p)  \, fpprec \,
1 1 1 16
2 1 1 16
3 3 2 16
4 6 2 16
5 15 7 16
6 27 13 16
7 63 34 32
8 120 67 64
9 252 144 128
10 495 286 300
11 1023 596 600



Here is Maxima code for gc\,

period_Max:11;
/* ----------------- definitions -------------------------------*/
/* basic function  = monic and centered complex quadratic polynomial 
http://en.wikipedia.org/wiki/Complex_quadratic_polynomial
*/
f(z,c):=z*z+c $
/* iterated function */
fn(n, z, c) :=
 if n=1         then f(z,c)
 else f(fn(n-1, z, c),c) $
/* roots of Fn are periodic point of  fn function */
Fn(n,z,c):=fn(n, z, c)-z $
/* gives irreducible divisors of polynomial Fn[p,z=0,c] */
GiveG[p]:=
block(
[f:divisors(p),t:1],
g,
f:delete(p,f),
if p=1 
then return(Fn(p,0,c)),
for i in f do t:t*GiveG[i],
g: Fn(p,0,c)/t,  
return(ratsimp(g))
 )$
/* degree of function */
GiveDegree(_function,_var):=hipow(expand(_function),_var);  
log10(x) := log(x)/log(10);
/* ------------------------------*/
file_output_append:true; /* to append new string to file */
grind:true;  
for period:1 thru period_Max step 1 do
(
g[period]:GiveG[period], /* function g */
d[period]:GiveDegree(g[period],c), /* degree of function g */
cf[period]:makelist(coeff(g[period],c,d[period]-i),i,0,d[period]), /* list of coefficients */
cf_max[period]:apply(max, cf[period]), /* max coefficient */
disp(cf_max[period]," ",ceiling(log10(cf_max[period]))),
s:concat("period:",string(period),"  cf_max:",cf_max[period]),
stringout("max_coeff.txt",s)/* save output to file as Maxima expressions */
);

Graphical methods for finding centers[edit]

All these methods shows centers for period n and its divisors.

Animation showing colors proportional to absolute value of iterations 1-20
Absolute value of first iteration
Color shows in which quadrant 5-th iteration lands
color is proportional to magnitude of zn[edit]
  • color is proportional to magnitude of zn[70]
  • Parameter plane is scanned for points c for which orbit of critical point vanishes [71]
  • youtube video : Mandelbrot Oscillations [72]
color shows in which quadrant zn lands[edit]
ninth-decomposition. Image and src code

This is radial nth-decomposition of exterior of Mandelbrot set ( compare it with n-th decomposition of LSM/M )

4 colors are used because there are 4 quadrants :

  • re(z_n) > 0 and im(z_n) > 0 ( 1-st quadrant )
  • re(z_n) < 0 and im(z_n) > 0 ( 2-nd quadrant )
  • re(z_n) < 0 and im(z_n) < 0 ( 3-rd quadrant )
  • re(z_n) > 0 and im(z_n) < 0 (4-th quadrant ).


"... when the four colors are meeting somewhere, you have a zero of q_n(c), i.e., a center of period dividing n. In addition, the light or dark color shows if c is in M." ( Wolf Jung )


Here is fragment of c code for computing 8-bit color for point c = cx + cy * i  :

/* initial value of orbit = critical point Z= 0 */
                       Zx=0.0;
                       Zy=0.0;
                       Zx2=Zx*Zx;
                       Zy2=Zy*Zy;
                       /* the same number of iterations for all points !!! */
                       for (Iteration=0;Iteration<IterationMax; Iteration++)
                       {
                           Zy=2*Zx*Zy + Cy;
                           Zx=Zx2-Zy2 +Cx;
                           Zx2=Zx*Zx;
                           Zy2=Zy*Zy;
                       };
                       /* compute  pixel color (8 bit = 1 byte) */
                       if ((Zx>0) && (Zy>0)) color[0]=0;   /* 1-st quadrant */
                       if ((Zx<0) && (Zy>0)) color[0]=10; /* 2-nd quadrant */
                       if ((Zx<0) && (Zy<0)) color[0]=20; /* 3-rd quadrant */
                       if ((Zx>0) && (Zy<0)) color[0]=30; /* 4-th quadrant */

One can also find cpp code in function quadrantqn from class mndlbrot in mndlbrot.cpp file from source code of program Mandel by Wolf Jung[73]

The differences from standard loop are :

  • there is no bailout test (for example for max value of abs(zn) ). It means that every point has the same number of iterations. For large n overflow is possible. Also one can't use test Iteration==IterationMax
  • Iteration limit is relatively small ( start for example IterationMax = 8 )

See also code in Sage[74]

using Gaussian wave function method [75][edit]

Video[edit]

One can make videos using :

  • going along some paths on parameter plane ( for example internal and external rays )
  • zoom into parameter plane[76][77][78] using automatic determination of Iteration Max number[79]
  • changing coloring scheme ( for example color cycling - Fractint)
  • changing some parameters of algorithm, for example :
    • maximal iteration of escape time algorithm
    • bailout value [80]

Speed improvements - optimisation[edit]

Symmetry[edit]

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 )[81]

Bailout test[edit]

Instead of checking :

 \sqrt{Z_x^2 + Z_y^2} < ER 

compute ER2 once and check :

 Z_x^2 + Z_y^2 < ER^2 

It gives the same result and is faster.

Period detection[edit]

"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 [82]


 // 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
                         }

Cardioid and period-2 checking[edit]

One way to improve calculations is to find out beforehand whether the given point lies within the cardioid or in the period-2 bulb. Before passing the complex value through the escape time algorithm, first check if:

 (x+1)^2 + y^2 < \frac{1}{16}

to determine if the point lies within the period-2 bulb and

 q = \left(x - \frac{1}{4}\right)^2 + y^2
 q \left(q + \left(x - \frac{1}{4}\right)\right) < \frac{1}{4}y^2.

to determine if the point lies inside the main cardioid.

Periodicity checking[edit]

Most points inside the Mandelbrot set oscillate within a fixed orbit. There could be anything from ten to tens of thousands of points in between, but if an orbit ever reaches a point where it has been before then it will continually follow this path, will never tend towards infinity and hence is in the Mandelbrot set. This Mandelbrot processor includes periodicity checking (and p-2 bulb/cardioid checking) for a great speed up during deep zoom animations with a high maximum iteration value.

More tutorials and code[edit]

References[edit]

  1. parameter plane in wikipedia
  2. Mandelbrot set in wikipedia
  3. complex quadratic polynomial in wikipedia
  4. reenigne blog : mandelbrot-set-taxonomy
  5. Mathematics of Divergent Fractals by
  6. wikipedia : Orbit trap
  7. Java program by Dieter Röß showing result of changing initial point of Mandelbrot iterations
  8. Fractal Benchmark by Erik Wrenholt
  9. The Computer Language Benchmarks Game
  10. LIsp Program by Frank Buss
  11. Mandelbrot Set ASCII art at Bill Clementson's blog
  12. mandel function from 2.5.11.14 Fractal Patterns at Pov-Ray docs
  13. Drawing the Mandelbrot set by the method of escape lines. M. Romera et al.
  14. http://www.metabit.org/~rfigura/figura-fractal/math.html boundary trace by Robert Figura
  15. http://web.archive.org/20010415125044/www.geocities.com/CapeCanaveral/Launchpad/5113/fr27.htm%7C An open letter to Dr. Meech from Joyce Haslam in FRACTAL REPORT 27
  16. mandelbrot set n-th-decomposition
  17. fractalforums : What range/precision for fractional escape counts for Mandelbrot/Julia sets?
  18. Making Mandelbrot Set Movies by Tony Finch
  19. Linas Vepstas. Renormalizing the mandelbrot escape.
  20. mpmath Python library
  21. | The Mandelbrot Function by John J. G. Savard
  22. Smooth colouring is the key to the Mandelbrot set by Tony Finch
  23. Wolf Jung explanation
  24. An algorithm to draw external rays of the Mandelbrot set by Tomoki Kawahira
  25. Wolf Jung's test for precision of drawing parameter rays
  26. Devaney In Global Analysis of Dynamical Systems, ed.: H. Broer, B. Krauskopf, G. Vegter. IOP Publishing (2001), 329-338 or Homoclinic Points in Complex Dynamical Systems
  27. mathoverflow : Parametrization of the boundary of the Mandelbrot set
  28. Boundary Scanning by Robert P. Munafo, 1993 Feb 3.
  29. The Almond Bread Homepage
  30. Efficient Boundary Tracking Through Sampling by Alex Chen , Todd Wittman , Alexander Tartakovsky , and Andrea Bertozzi
  31. http://www.robertnz.net/cx.htm contour integration by Robert Davies
  32. Combinatorics in the Mandelbrot Set - Lavaurs Algorithm by
  33. Lavaurs algorithm by Michael Frame, Benoit Mandelbrot, and Nial Neger in lisp by Ruben Berenguel
  34. Abstract Mandelbrot tree by claudius maximus
  35. The Mandelbrot cactus by Robert L. Devaney
  36. Burns A M : : Plotting the escape- An animation of parabolic bifurrcation in the mandelbrot set. Mathematics Magazine: Volume 75, Number 2, Pages: 104-116 page 104
  37. VISUALIZATION OF THE BIFURCATION LOCUS OF CUBICPOLYNOMIAL FAMILY by HIROYUKI INOU
  38. MIsiurewicz point in wikipedia
  39. Wolf Jung
  40. The logistic equation by Didier Gonze October 4, 2013
  41. Ljapunov Exponent and mandelbrot set by janthor
  42. Image by Anders Sandberg
  43. Fractint : Misc. Options and algorithms
  44. Java™ Number Cruncher: The Java Programmer's Guide to Numerical Computing By Ronald Mak
  45. Firefly Application Help by Terry W. Gintz
  46. Fractint doc by Noel Giffin
  47. gnofract4d
  48. Fractint doc by Noel Giffin
  49. A Series of spiral bays in the Mandelbrot set by Patrick Hahn
  50. gnofract4d
  51. Rigorous Investigations Of Periodic Orbits In An Electronic Circuit By Means Of Interval Methods by Zbigniew Galias
  52. Mandelbrot set drawing by Milan
  53. The Mandelbrot set by Renato Fonseca
  54. Mandelbrot set Components : image with descriptions and references
  55. Young Hee Geum, Kevin G. Hare : Groebner Basis, Resultants and the generalized Mandelbrot Set - methods for period 2 and higher, using factorization for finding irreducible polynomials
  56. Image : parametrisation of boundary with Newton method near centers of componentswith src code
  57. Mark McClure "Bifurcation sets and critical curves" - Mathematica in Education and Research, Volume 11, issue 1 (2006).
  58. Burns A M : Plotting the Escape: An Animation of Parabolic Bifurcations in the Mandelbrot Set. Mathematics Magazine, Vol. 75, No. 2 (Apr., 2002), pp. 104-116
  59. mathoverflow : what-are-good-methods-for-detecting-parabolic-components-and-siegel-disk-componen by Lasse Rempe-Gillen
  60. Newsgroups: gmane.comp.mathematics.maxima.general Subject: system of equations Date: 2008-08-11 21:44:39 GMT
  61. Thayer Watkins : The Structure of the Mandelbrot Set
  62. Brown Method by Robert P. Munafo
  63. Mandel: software for real and complex dynamics by Wolf Jung
  64. Piers Lawrence and Rob Corless : Mandelbrot Polynomials and Matrices. 1 ; 048 ; 575 roots of period 21
  65. [http://www.iec.csic.es/~gerardo/publica/Alvarez98a.pdf Determination of Mandelbrot Set's Hyperbolic Component Centres by G Alvarez, M Romera, G Pastor and F Montoya. Chaos, Solitons & Fractals 1998, Vol 9 No 12 pp 1997-2005]
  66. Myrberg, P J, Iteration der reelen polynome zweiten GradesIII, Ann. Acad. Sci. Fennicae, A. I no. 348, 1964.
  67. evaldraw script that can compute 1 million root by knighty
  68. irreducible divisors at wikipedia
  69. V Dolotin , A Morozow : On the shapes of elementary domains or why Mandelbrot set is made from almost ideal circles ?
  70. [http://www.cerebralmastication.com/2009/02/mandelbrot-set-in-r/ Mandelbrot Set in R by J.D. Long]
  71. Fractal Stream help file
  72. youtube video : Mandelbrot Oscillations
  73. Multiplatform cpp program Mandel by Wolf Jung
  74. Formation of Escape-Time Fractals By Christopher Olah
  75. [http://www.dhushara.com/DarkHeart/DarkHeart.htm#_3__The_Dark Exploding the Dark Heart of Chaos by Chris King March-Dec 2009]
  76. Really Deep Fractal Zoom Movie – Much Faster by Bruce Dawson
  77. Making Mandelbrot Set Movies by Tony Finch
  78. MLbrot by Daniel de Rauglaud
  79. Discussion : A way to determine the ideal number of maximum iterations for an arbitrary zoom level in a Mandelbrot fractal
  80. Gif image by jgabase : a wormhole effect on your fractals by changing the bailout dynamicaly
  81. How to use symetry of set
  82. FractalNet by Michael Hogg
  83. ASCII graphic

Notes[edit]

  1. ^ It is conjectured that the polynomials (for the centers) are irreducible, but this has not been proved yet - Wolf Jung