Fractals/Iterations in the complex plane/Julia set

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This book shows how to code different algorithms for drawing sets in dynamical plane : Julia, Filled-in Julia or Fatou sets for complex quadratic polynomial. It is divided in 2 parts :

  • description of various algorithms[1]
  • descriptions of technics for visualisation of various sets in dynamic plane
    • Julia set
    • Fatou set
      • basin of attraction of infinity ( open set)
      • basin of attraction of finite attractor

Various types of dynamics needs various algorithms



Methods based on speed of attraction[edit]

Here color is proportional to speed of attraction ( convergence to attractor). These methods are used in Fatou set.

Basin of attraction to infinity = exterior of filled-in Julia set and The Divergence Scheme = Escape Time Method ( ETM )[edit]

First read definitions

Here one computes forward iterations of a complex point Z0:

Here is function which computes the last iteration, that is the first iteration that lands in the target set ( for example leaves a circle around the origin with a given escape radius ER ) for the iteration of the complex quadratic polynomial above. It is a iteration ( integer) for which (abs(Z)>ER). It can also be improved [2]

C version ( here ER2=ER*ER) using double floating point numbers ( without complex type numbers) :

 int GiveLastIteration(double Zx, double Zy, double Cx, double Cy, int IterationMax, int ER2)
  double Zx2, Zy2; /* Zx2=Zx*Zx;  Zy2=Zy*Zy  */
  int i=0;
  while (i<IterationMax && (Zx2+Zy2<ER2) ) /* ER2=ER*ER */
   Zy=2*Zx*Zy + Cy;
   Zx=Zx2-Zy2 +Cx;
  return i;

C with complex type from GSL :[3]

#include <gsl/gsl_complex.h>
#include <gsl/gsl_complex_math.h>
#include <stdio.h>
// gcc -L/usr/lib -lgsl -lgslcblas -lm t.c 
// function fc(z) = z*z+c

gsl_complex f(gsl_complex z, gsl_complex c) {
  return gsl_complex_add(c, gsl_complex_mul(z,z));

int main () {
  gsl_complex c = gsl_complex_rect(0.123, 0.125);
  gsl_complex z = gsl_complex_rect(0.0, 0.0);
  int i;
  for (i = 0; i < 10; i++) {
    z = f(z, c);
    double zx = GSL_REAL(z);
    double zy = GSL_IMAG(z);
    printf("Real: %f4 Imag: %f4\n", zx, zy);
  return 0;

C++ versions:

 int GiveLastIteration(complex C,complex Z , int imax, int ER)
   int i; // iteration number
   for(i=0;i<=imax-1;i++) // forward iteration
      Z=Z*Z+C; // overloading of operators
      if(abs(Z)>ER) break;
   return i;

#include <complex> // C++ complex library 

// bailout2 = bailout * bailout
// this function is based on function esctime from mndlbrot.cpp 
// from program mandel ver. 5.3 by Wolf Jung

int escape_time(complex<double> Z, complex<double> C , int iter_max,  double bailout2)
  // z= x+ y*i   z0=0
  long double x =Z.real(), y =Z.imag(),  u ,  v ;
  int iter; // iteration
  for ( iter = 0; iter <= iter_max-1; iter++)
  { u = x*x; 
    v = y*y;
    if ( u + v <= bailout2 ) 
         y = 2 * x * y + C.imag();
         x = u - v + C.real(); 
       } // if
    else break;
  } // for 
  return iter;
} // escape_time


Delphi version ( using user defined complex type, cabs and f functions )

function GiveLastIteration(z,c:Complex;ER:real;iMax:integer):integer;
  var i:integer;
  while (cabs(z)<ER) and (i<iMax) do
      z:= f(z,c);
  result := i;

where :

type complex = record x, y: real; end;

function cabs(z:complex):real;

function f(z,c:complex):complex; // complex quadratic polynomial
  var tmp:complex;
  tmp.x := (z.x*z.x) - (z.y*z.y) + c.x;
  tmp.y := 2*z.x*z.y + c.y ;
  result := tmp;


Delphi version without explicit definition of complex numbers :

function GiveLastIteration(zx0,zy0,cx,cy,ER2:extended;iMax:integer):integer;
  // iteration of z=zx+zy*i under fc(z)=z*z+c
  // where c=cx+cy*i
  // until abs(z)<ER  ( ER2=ER*ER )  or i>=iMax
  var i:integer;

  while (zx2+zy2<ER2) and (i<iMax) do
      zy:=2*zx*zy + cy;
      zx:=zx2-zy2 +cx;
  result := i;

Euler version by R. Grothmann ( with small change : from z^2-c to z^2+c) [5]

function iter (z,c,n=100) ...

loop 1 to n;
h=h^2 + c;
if totalmax(abs(h))>1e20; m=#; break; endif;
return {h,m};

Lisp version

This version uses complex numbers. It makes the code short but is also inefficien.

  (SETQ Z Z_0) 
  (SETQ I 0)
    (INCF I)
    (SETQ Z (+ (* Z Z) _C)))

Maxima version :

/* easy to read but very slow version, uses complex type numbers */ 
  while abs(z)<ER and i<iMax
     do (z:z*z + c,i:i+1),

/* faster version, without use of complex type numbers,
   compare with c version, ER2=ER*ER */  
 while (zx2+zy2<ER2) and i<iMax do
  zy:2*zx*zy + cy,
  zx:zx2-zy2 +cx,

Boolean Escape time[edit]

Algorithm: for every point z of dynamical plane (z-plane) compute iteration number ( last iteration) for which magnitude of z is greater than escape radius. If last_iteration=max_iteration then point is in filled-in Julia set, else it is in its complement (attractive basin of infinity ). Here one has 2 options, so it is named boolean algorithm.

if (LastIteration==IterationMax)
   then color=BLACK;   /* bounded orbits = Filled-in Julia set */
   else color=WHITE;  /* unbounded orbits = exterior of Filled-in Julia set  */

In theory this method is for drawing Filled-in Julia set and its complement ( exterior), but when c is Misiurewicz point ( Filled-in Julia set has no interior) this method draws nothing. For example for c=i . It means that it is good for drawing interior of Filled-in Julia set .

ASCII graphic[edit]
; common lisp
(loop for y from -2 to 2 by 0.05 do
      (loop for x from -2 to 2 by 0.025 do
		(let* ((z (complex x y))
                   	(c (complex -1 0))
                   	(iMax 20)
			(i 0))

		(loop  	while (< i iMax ) do 
			(setq z (+ (* z z) c))
			(incf i)
			(when (> (abs z) 2) (return i)))

           (if (= i iMax) (princ (code-char 42)) (princ (code-char 32)))))
      (format t "~%"))
PPM file with raster graphic[edit]
Filled-in Julia set for c= =-1+0.1*i. Image and C source code

Integer escape time = Level Sets of the Basin of Attraction of Infinity = Level Sets Method= LSM/J[edit]

level sets of escape time for C=0 and 0.9<Z<1.5

Escape time measures time of escaping to infinity ( infinity is superattracting point for polynomials). Time is measured in steps ( iterations = i) needed to escape from circle of given radius ( ER= Escape Radius).

One can see few things:

Level sets here are sets of points with the same escape time. Here is algorithm of choosing color in black & white version.

  if (LastIteration==IterationMax)
   then color=BLACK;   /* bounded orbits = Filled-in Julia set */
   else   /* unbounded orbits = exterior of Filled-in Julia set  */
        if ((LastIteration%2)==0) /* odd number */
           then color=BLACK; 
           else color=WHITE;

Normalized iteration count (real escape time or fractional iteration)[edit]

Math formula :

Maxima version :

/* */
 while abs(z)<E_R and i<i_Max
   do (z:z*z + c,i:i+1),

In Maxima log(x) is a natural (base e) logarithm of x. To compute log2 use :

log2(x) := log(x) / log(2);

Level Curves of escape time Method = eLCM/J[edit]

These curves are boundaries of Level Sets of escape time ( eLSM/J ). They can be drawn using these methods:

  • edge detection of Level Curves ( =boundaries of Level sets).
    • Algorithm based on paper by M. Romera et al.[6]
    • Sobel filter
  • drawing lemniscates = curves , see explanation and source code
  • drawing circle and its preimages. See this image, explanation and source code
  • method described by Harold V. McIntosh[7]
/* Maxima code : draws lemniscates of Julia set */
c: 1*%i;
f[n](z) := if n=0 then z else (f[n-1](z)^2 + c);
load(implicit_plot); /* package by Andrej Vodopivec */ 

Basin of attraction of finite attractor = interior of filled-in Julia set[edit]

  • How to find periodic attractor ?
  • How many iterations is needed to reach attractor ?
Distance between points and iteration

Components of Interior of Filled Julia set ( Fatou set)[edit]

  • use limited color ( palette = list of numbered colors)
  • find period of attracting cycle
  • find one point of attracting cycle
  • compute number of iteration after when point reaches the attractor
  • color of component=iteration % period[8]
  • use edge detection for drawing Julia set

Internal Level Sets[edit]

See :

  • algorithm 0 of program Mandel by Wolf Jung

Decomposition of target set[edit]

Binary decomposition[edit]

Here color of pixel ( exterior of Julia set) is proportional to sign of imaginary part of last iteration .

Main loop is the same as in escape time.

In other words target set is decompositioned in 2 parts ( binary decomposition) :

Algorithm in pseudocode ( Im(Zn) = Zy ) :

if (LastIteration==IterationMax)
   then color=BLACK;   /* bounded orbits = Filled-in Julia set */
   else   /* unbounded orbits = exterior of Filled-in Julia set  */
        if (Zy>0) /* Zy=Im(Z) */
           then color=BLACK; 
           else color=WHITE;

Modified decomposition[edit]

Here exterior of Julia set is decompositioned into radial level sets.

It is because main loop is without bailout test and number of iterations ( iteration max) is constant.

It creates radial level sets. See also video by bryceguy72[9] and by FreymanArt[10]

  for (Iteration=0;Iteration<8;Iteration++)
 /* modified loop without checking of abs(zn) and low iteration max */
    Zy=2*Zx*Zy + Cy;
    Zx=Zx2-Zy2 +Cx;
  /* --------------- compute  pixel color (24 bit = 3 bajts) */
 /* exterior of Filled-in Julia set  */
 /* binary decomposition  */
  if (Zy>0 ) 
    array[iTemp]=255; /* Red*/
    array[iTemp+1]=255;  /* Green */ 
    array[iTemp+2]=255;/* Blue */
  if (Zy<0 )
    array[iTemp]=0; /* Red*/
    array[iTemp+1]=0;  /* Green */ 
    array[iTemp+2]=0;/* Blue */    

It is also related with automorphic function for the group of Mobius transformations [11]

Inverse Iteration Method (IIM/J) : Julia set[edit]

Inverse iteration of repellor for drawing Julia set

Complex potential - Boettcher coordinate[edit]

See description here


This algorithm is used when dynamical plane consist of two of more basins of attraction. For example for c=0.

It is not appropiate when interior of filled Julia set is empty, for example for c=i.

Description of algorithm :

  • for every pixel of dynamical plane do :
    • compute 4 corners ( vertices) of pixel ( where lt denotes left top, rb denotes right bottom, ... )
    • check to which basin corner belongs ( standard escape time and bailout test )
    • if corners do not belong to the same basin mark it as Julia set

Examples of code

  • program in Pascal[12]
  • via convolution with a kernel [13]


This algorith has 2 versions :

  • exterior
  • interior

Compare it with version for parameter plane and Mandelbrot set : DEM/J

Internal distance estimation[edit]

External distance estimation[edit]

" For distance estimate it has been proved that the computed value differs from the true distance at most by a factor of 4. " (Wolf Jung)

Math formula :

where :

is first derivative with respect to c.

This derivative can be found by iteration starting with

and then :

How to use distance[edit]

One can use distance for colouring  :

  • only Julia set ( boundary of filled Julia set)
  • boundary and exterior of filled Julia set.

Here is first example :

if (LastIteration==IterationMax) 
    then { /*  interior of Julia set, distance = 0 , color black */ }
    else /* exterior or boundary of Filled-in Julia set  */
         {  double distance=give_distance(Z0,C,IterationMax);
            if (distance<distanceMax)
                then { /*  Julia set : color = white */ }
                else  { /*  exterior of Julia set : color = black */}

Here is second example [14]

if (LastIteration==IterationMax) or distance < distanceMax then ... // interior by ETM/J and boundary by DEM/J
else .... // exterior by real escape time


DistanceMax is smaller than pixel size. During zooming pixel size is decreasing and DistanceMax should also be decreased to obtain good picture. It can be made by using formula :


One can start with n=1 and increase n if picture is not good. Check also iMax !!

DistanceMax may also be proportional to zoom factor [15] :

where thick is image width ( in world units) and mag is a zoom factor.

Examples of code[edit]

For cpp example see mndlbrot::dist from mndlbrot.cpp in src code of program mandel ver 5.3 by Wolf Jung.

C function :

/*based on function  mndlbrot::dist  from  mndlbrot.cpp
 from program mandel by Wolf Jung (GNU GPL )  */
double jdist(double Zx, double Zy, double Cx, double Cy ,  int iter_max)
 int i;
 double x = Zx, /* Z = x+y*i */
         y = Zy, 
         /* Zp = xp+yp*1 = 1  */
         xp = 1, 
         yp = 0, 
         /* temporary */
         /* a = abs(z) */
 for (i = 1; i <= iter_max; i++)
  { /* first derivative   zp = 2*z*zp  = xp + yp*i; */
    nz = 2*(x*xp - y*yp) ; 
    yp = 2*(x*yp + y*xp); 
    xp = nz;
    /* z = z*z + c = x+y*i */
    nz = x*x - y*y + Cx; 
    y = 2*x*y + Cy; 
    x = nz; 
    /* */
    nz = x*x + y*y; 
    nzp = xp*xp + yp*yp;
    if (nzp > 1e60 || nz > 1e60) break;
 /* distance = 2 * |Zn| * log|Zn| / |dZn| */
 return 2* a*log(a)/sqrt(nzp); 

Delphi function :

function Give_eDistance(zx0,zy0,cx,cy,ER2:extended;iMax:integer):extended;

var zx,zy ,  // z=zx+zy*i
    dx,dy,  //d=dx+dy*i  derivative :  d(n+1)=  2 * zn * dn
    z2,  //
    d2, //
    a // abs(d2)
  //initial values
  // d0=1
  // to remove warning : variables may be not initialised ?

  for i := 0 to iMax - 1 do
      // first derivative   d(n+1) = 2*zn*dn  = dx + dy*i;
      dx_temp := 2*(zx*dx - zy*dy) ;
      dy := 2*(zx*dy + zy*dx);
      dx := dx_temp;
      // z = z*z + c = zx+zy*i
      zx_temp := zx*zx - zy*zy + Cx;
      zy := 2*zx*zy + Cy;
      zx := zx_temp;
      if ((z2>1e60) or (d2 > 1e60)) then  break;
    end; // for i
   if (d2 < 0.01) or (z2 < 0.1)  // when do not use escape time
    then  result := 10.0
        // distance = 2 * |Zn| * log|Zn| / |dZn|
        result := 2* a*log10(a)/sqrt(d2);

end;  //  function Give_eDistance

Matlab code by Jonas Lundgren[16]

function D = jdist(x0,y0,c,iter,D)
%JDIST Estimate distances to Julia set by potential function
% by Jonas Lundgren
% Code covered by the BSD License

% Escape radius^2
R2 = 100^2;

% Parameters
N = numel(x0);
M = numel(y0);
cx = real(c);
cy = imag(c);
iter = round(1000*iter);

% Create waitbar
h = waitbar(0,'Please wait...','name','Julia Distance Estimation');
t1 = 1;

% Loop over pixels
for k = 1:N/2
    x0k = x0(k);
    for j = 1:M
        % Update distance?
        if D(j,k) == 0
            % Start values
            n = 0;
            x = x0k;
            y = y0(j);
            b2 = 1;                 % |dz0/dz0|^2
            a2 = x*x + y*y;         % |z0|^2
            % Iterate zn = zm^2 + c, m = n-1
            while n < iter && a2 <= R2
                n = n + 1;
                yn = 2*x*y + cy;
                x = x*x - y*y + cx;
                y = yn;
                b2 = 4*a2*b2;       % |dzn/dz0|^2
                a2 = x*x + y*y;     % |zn|^2
            % Distance estimate
            if n < iter
                % log(|zn|)*|zn|/|dzn/dz0|
                D(j,k) = 0.5*log(a2)*sqrt(a2/b2);
    % Lap time
    t = toc;
    % Update waitbar
    if t >= t1
        str = sprintf('%0.0f%% done in %0.0f sec',200*k/N,t);
        t1 = t1 + 1;

% Close waitbar

Maxima function :

/* needs z in exterior of Kc */
cabsdz:1, /* overflow limit */
while  cabsdz < 10000000 and  i<i_max
   z:z*z + c,


In this algorithm distances between 2 points of the same orbit are checked

average discrete velocity of orbit[edit]

average discrete velocity of orbit - code and description

It is used in case of :

Cauchy Convergence Algorithm (CCA)[edit]

This algorithm is described by User:Georg-Johann. Here is also Matemathics code by Paul Nylander


Normality In this algorithm distances between points of 2 orbits are checked

Checking equicontinuity by Michael Becker[edit]

"Iteration is equicontinuous on the Fatou set and not on the Julia set". (Wolf Jung) [17][18]

Michael Becker compares the distance of two close points under iteration on Riemann sphere.[19] [20]

This method can be used to draw not only Julia sets for polynomials ( where infinity is always superattracting fixed point) but it can be also applied to other functions ( maps), for which infinity is not an attracting fixed point.[21]

using Marty's criterion by Wolf Jung[edit]

Wolf Jung is using "an alternative method of checking normality, which is based on Marty's criterion: |f'| / (1 + |f|^2) must be bounded for all iterates." It is implemented in mndlbrot::marty function ( see src code of program Mandel ver 5.3 ). It uses one point of dynamic plane.

Koenigs coordinate[edit]

Koenigs coordinate are used in the basin of attraction of finite attracting (not superattracting) point (cycle).



You don't need a square root to compare distances. [22]


Julia sets can have many symmetries [23][24]

Quadratic Julia set has allways rotational symmetry ( 180 degrees) :

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

when c is on real axis ( cy = 0) Julia set is also reflection symmetric :[25]

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

Algorithm :

  • compute half image
  • rotate and add the other half
  • write image to file [26]



Target set[edit]

Target set or trap

One can divide it according to :

  • attractors ( finite or infinite)
  • dynamics ( hyperbolic, parabolic, elliptic )

For infinite attractor - hyperbolic case[edit]

Target set is an arbitrary set on dynamical plane containing infinity and not containing points of Filled-in Fatou sets.

For escape time algorithms target set determines the shape of level sets and curves. It does not do it for other methods.

Exterior of circle[edit]

This is typical target set. It is exterior of circle with center at origin and radius =ER :

Radius is named escape radius ( ER ) or bailout value. Radius should be greater than 2.

Exterior of square[edit]

Here target set is exterior of square of side length centered at origin

For finite attractors[edit]

internal level sets around fixed point

See :

Julia sets[edit]

"Most programs for computing Julia sets work well when the underlying dynamics is hyperbolic but experience an exponential slowdown in the parabolic case." ( Mark Braverman )[28]

  • when Julia set is a set of points that do not escape to infinity under iteration of the quadratic map ( = filled Julia set has no interior = dendrt)
    • IIM/J
    • DEM/J
    • checking normality
  • when Julia set is a boundary between 2 basin of attraction ( = filled Julia set has no empty interior) :
    • boundary scaning [29]
    • edge detection

Fatou set[edit]

Interior of filled Julia set can be coloured :

Periodic points[edit]

More is here


One can make videos using :

  • zoom into dynamic plane
  • changing parametr c along path inside parameter plane[32]
  • changing coloring scheme ( for example color cycling )

Examples :

More tutorials and code[edit]


  1. Standard coloring algorithms from Ultra Fractal
  2. Faster Fractals Through Algebra by Bruce Dawson ( author of Fractal eXtreme)
  3. C code with gsl from tensorpudding
  4. Program Mandel by Wolf Jung on GNU General Public License
  5. Euler examples by R. Grothmann
  6. Drawing the Mandelbrot set by the method of escape lines. M. Romera et al.
  7. Julia Curves, Mandelbrot Set, Harold V. McIntosh.
  8. The fixed points and periodic orbits by E Demidov
  9. Video : Julia Set Morphing with Magnetic Field lines by bryceguy72
  10. Video : Mophing Julia set with color bands / stripes by FreymanArt
  11. Gerard Westendorp : Platonic tilings of Riemann surfaces - 8 times iterated Automorphic function z->z^2 -0.1+ 0.75i
  12. Pascal program fo BSM/J by Morris W. Firebaugh
  13. Boundary scanning and complex dynamics by Mark McClure
  14. Pictures of Julia and Mandelbrot sets by Gert Buschmann
  15. Pictures of Julia and Mandelbrot sets by Gert Buschmann
  16. Julia sets by Jonas Lundgren in Matlab
  17. Alan F. Beardon, S. Axler, F.W. Gehring, K.A. Ribet : Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, 2000; ISBN 0387951512, 9780387951515; page 49
  18. Joseph H. Silverman : The arithmetic of dynamical systems. Springer, 2007. ISBN 0387699031, 9780387699035; page 22
  19. Visualising Julia sets by Georg-Johann
  20. Problem : How changes distance between 2 near points under iteration ? Can I tell to which set these points belong when I know it ?
  21. Julia sets by Michael Becker. See the metric d(z,w)
  22. Algorithms : Distance_approximations in wikibooks
  23. The Julia sets symmetry by Evgeny Demidov
  24. mathoverflow : symmetries-of-the-julia-sets-for-z2c
  25. htJulia Jewels: An Exploration of Julia Sets by Michael McGoodwin (March 2000)
  26. julia sets in Matlab by Jonas Lundgren
  27. Tessellation of the Interior of Filled Julia Sets by Tomoki Kawahira
  28. Mark Braverman : On efficient computation of parabolic Julia sets
  30. Ray Tracing Quaternion Julia Sets on the GPU by Keenan Crane
  31. Tessellation of the Interior of Filled Julia Sets by Tomoki Kawahira
  32. Julia-Set-Animations at devianart