Fractals/Iterations in the complex plane/Mandelbrot set

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

Exterior of Mandelbrot set

Colouring of exterior of Mandelbrot set can be :

• boolean ( bETM/M )
• discrete ( LSM/M = iETM/M)
• Smooth  :
  • Real Escape Time ( rETM/M )
  • Distance estimation ( DEM/M )
  • Complex potential ( CPM/M )

One can also draw curves :

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

Escape time

Here for given point c one checks how critical point z₀ = 0.0 behaves on dynamical plane . If you change initial point ypu will get different result

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

Boolean escape time

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)

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

Graphic file ( PPM )

Here are various programs for creating pbm file

C

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

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

• 

  Number of details is proportional to maximal number of iterations

• 

  Mandelbrot animation based on a static number of iterations per pixel. Here you can see why offset is sometimes used ( because - color gradient changes : for high MaxIteration disapears.

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

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++

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

//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

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

Whole Lisp program making ASCI graphic based on code by Frank Buss

; 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

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

Pov-Ray has a built-in function mandel

Matemathica

Here is code by Paul Nylander

Level Curves of escape time Method = LCM/M

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. This method is fast and allows looking for high iterations.
• boundary trace
• 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

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

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.

nth-decomposition

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).

Real Escape Time

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 :

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

where :

• log(log(bailout) can be precaclulated

C

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);

Here is another version by Tony Finch

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

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

C++

// 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

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

cet = n + log₂ln(R) − log₂ln | z |

Python

Python code using mpmath library

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

Exterior DEM/M

simple boundary with DEM/M

Boundary with DEM/M and Sobel filter

Estimates distance from point $c\,$ ( in the exterior of Mandelbrot set ) to nearest point in Mandelbrot set.

Math formula :

$distance(z_0,\sigma M) =\lim_{n \to \infty} 2 \frac{|z_n|}{|z'_n|}\ln|z_n|$

It can be used to create B&W images from BOF :

• map 41 on page 84
• map 43 on page 85
• an unnumbered plot on page 188

Creating DEM images can be improved by use of :

• the Koebe 1/4-theorem
• gray scale based on distance
• Fractal forum : Relationship between bailout and accuracy of analytical DE

Two algorithm in two loops

Here is cpp function. It gives integer index of color as an output.

// this function is  from program mandel ver 5.3 by Wolf Jung
// http://www.iram.rwth-aachen.de/~jung/indexp.html 
// it needs escape time before
 
 
int mndlbrot::dist(double a, double b, double x, double y)
{  uint j; double xp = 1, yp = 0, nz, nzp; //zp = 1
   for (j = 1; j <= maxiter; j++)
   {  nz = 2*(x*xp - y*yp); yp = 2*(x*yp + y*xp); xp = nz; //zp = 2*z*zp;
      if (sign > 0) xp++; //zp = 2*z*zp + 1
      nz = x*x - y*y + a; y = 2*x*y + b; x = nz; //z = z*z + c;
      nz = x*x + y*y; nzp = xp*xp + yp*yp;
      if (nzp > 1e60 || nz > 1e60) break;
   }
   if (nz < bailout) return 1; //not escaping
   if (nzp < 0.01) return 11; //escaping through 0
   x = log(nz); x = x*x*nz / (temp[1]*nzp); //square of dist/pixelwidth
   if (x < 0.08) return 1;
   if (x < 0.15) return 9;
   if (x < 1.0) return 3;
   return 11;
} //dist

Two algorithms in one loop

Here is Java function. It computes in one loop both : iteration $z_n\,$ and derivative $z'_n\,$. If (on dynamic plane) critical point :

• does not escapes to infinity ( bailouts), then it belongs to interior of Mandelbrot set and has colour maxColor,
• escapes then it maybe in exterior or near boundary. Its colour is "proportional" to distance between the point c and the nearest point in the Mandelbrot set. It uses also colour cycling ( (int)R % maxColor ).

// Java function by Evgeny Demidov from http://www.ibiblio.org/e-notes/MSet/DEstim.htm
// based on code by Robert P. Munafo from http://www.mrob.com/pub/muency/distanceestimator.html
  public int iterate(double cr, double ci, double K, double f) {
    double Cr,Ci, I=0, R=0,  I2=I*I, R2=R*R, Dr=0,Di=0,D;   int n=0;
    if (f == 0){ Cr = cr; Ci = ci;}
    else{ Cr = ci; Ci = cr;}
    do  {
      D = 2*(R*Dr - I*Di) +1;  Di = 2*(R*Di + I*Dr);  Dr = D;
      I=(R+R)*I+Ci;  R=R2-I2+Cr;  R2=R*R;  I2=I*I;  n++;
    } while ((R2+I2 < 100.) && (n < MaxIt) );
    if (n == MaxIt) return maxColor; // interior
    else{ // boundary and exterior 
     R = -K*Math.log(Math.log(R2+I2)*Math.sqrt((R2+I2)/(Dr*Dr+Di*Di))); // compute distance
     if (R < 0) R=0;
     return (int)R % maxColor; }; 
 }

Complex potential

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.

Real potential = CPM/M

In Fractint :

potential =  log(modulus)/2^iterations

One can use real potential to:

• smooth (continuous) coloring
• 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

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

• • binary decomposition of LSM/M (see above)
  • nth-decomposition : color of exterior is proportional to quadrant in which last iteration lands. ( see below )

Then one go futher :

• Wolf Jung explanation
• Tomoki Kawahira paper

Tests

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.

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

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

Drawing boundaries

Methods used to draw boundary of Mandelbrot set :

• DEM/M
• Boundary scanning method (BSM) : detecting the edges after Boolean escape time
• boundary tracing method
• contour integration by Robert Davies
• Jungreis function $\Psi_M\,$
• drawing boundaries of components of Mandelbrot set
• Newton method

Topological models of Mandelbrot set

Topological model of Mandelbrot set cactus

Arcs up to period 12

• Thurston model ( abstract Mandelbrot set )
  • Lavaurs algorithm
  • Abstract Mandelbrot tree
• Cactus model ( 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

Computing Misiurewicz points of complex quadratic mapping

First method

Misiurewicz points 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

" 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."

Interior of Mandelbrot set - hyperbolic components

The Ljapunov Exponent

• image and description by janthor
• image by Anders Sandberg

Interior distance estimation

Interior distance estimation

Description of method

absolute value of the orbit

internal level sets

Color of point  :

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

bof60

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
• the smallest value z gets during iteration
• 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

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

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

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) .

• fragment of code : fractint.cfrm from 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
}

Period of hyperbolic components

period of hyperbolic components

Algorithms :

• direct period detection from iterations
• the spider algorithm
• "quick and dirty" algorithm ( check if abs(z_n) < eps then colour c-point with colour n )

Multiplier map

Internal rays

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.

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

Boundaries of hyperbolic components

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
  • using resultants
• parametrisation of boundary with Newton method near centers of components . 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.

solving boundary equations

System of 2 equations defining boundaries of period $n\,$ hyperbolic components

• first defines periodic point,
• second defines indifferent orbit.

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

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  :

$\lambda = w \,$

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
  • 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

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

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

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 which is based on paper of Wolf Jung.

Boundary equation

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

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

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

Centers of components

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
      • finding all roots of polynomial
      • Myrberg's method
• one center $c_p\,$ for given period $p\,$ near given point $c\,$
  • some steps of Newton method for solving $F_p(0,c_p) = 0 \,$

System of equations defining centers

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

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

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 c² + c = 0
• for period 3 : ((z^2+c)^2 +c)^2 +c = z and z=0 so (c² + 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

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 :

• $G\,$ is irreducible divisors of $F_p \,$
• capital Pi notation of iterated multiplication is used
• $m|p\,$ means here : for all divisors of $p\,$ ( from 1 to p ). See table of divisors.

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

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

Parameter plane is scanned for points c for which orbit of critical point vanishes

color shows in which quadrant zn lands

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

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

Video

One can make videos using :

• zoom into parameter plane using automatic determination of Iteration Max number
• changing coloring scheme ( for example color cycling - Fractint)
• changing some parameters of algorithm, for example maximal iteration of escape time algorithm

Speed improvements - optimisation

Speed improvements by Robert Munafo

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 )

Bailout test

Instead of checking :

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

check :

$Z_x^2 + Z_y^2 < ER^2$

It gives the same result and is faster.

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

 // 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

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

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

• in Basic see Mandelbrot Dazibao
• in Java see Evgeny Demidov
• in C see Linas Vepstas
• in C++ see Wolf Jung page,
• in Factor program by Slava Pestov
• in Gnuplot see Tutorial by T.Kawano
• in Lisp (Maxima) see :
  • Dynamics by Jaime E. Villate
  • Mu-Ency - The Encyclopedia of the Mandelbrot Set by Robert P. Munafo
  • Frac by Yannick Gingras with beautiful [2]
  • Fractals in Lisp by Takaya Iwamoto
• in Octave see wikibooks or another verion by Christopher Wellons

• How to Plot the Mandelbrot Set By Hand
• comparison of various languages :
  • Computer Language Benchmarks Game : Compare the performance of ≈30 programming languages using ≈12 flawed benchmarks for 4 different combinations of OS/machine.
  • rosettacode
  • Fractal Benchmark in Ruby, Io, PHP, Python, Lua, Java, Perl, Applescript, TCL, ELisp, Javascript, OCaml, Ghostscript, and C by Erik Wrenholt
  • in PDL, IDL, MATLAB, Octave, C and FORTRAN77 by Xavier Calbet
• ASCI graphic :
  • ASCII Graphics at Mu-Ency by Robert P. Munafo
  • using scripting languages by Warp
  • Fractal Benchmark by Theo Wollenleben
  • using Lisp on Bill Clementson's blog
• 3D
  • 3D images of Mandelbrot set
  • 3d images by Jos Leys

• 

  This is a 3D version of the image "Mandel zoom 03 seehorse.jpg" DEM

• 

  Half 3D hypercomplex Mandelbrot Fractal. Rendered with Mandelbulber 0.80.

Notes

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