Fractals/Iterations in the complex plane/Julia set

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 attractionEdit

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

How to find:

  • lowest optimal bailout values ( IterationMax) ? [2]
  • escape radius [3]

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 [4]

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 :[5]

#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) [7]

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 inefficient.

  (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 timeEdit

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 graphicEdit
; 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 graphicEdit
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/JEdit

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;

Here is the c function which:

  • uses complex double type numbers
  • computes 8 bit color ( shades of gray)
  • checks both escape and attraction test
unsigned char ComputeColorOfLSM(complex double z){

 int nMax = 255;
  double cabsz;
  unsigned char iColor;
  int n;

  for (n=0; n < nMax; n++){ //forward iteration
	cabsz = cabs(z);
    	if (cabsz > ER) break; // esacping
    	if (cabsz< PixelWidth) break; // fails into finite attractor = interior
     	z = z*z +c ; /* forward iteration : complex quadratic polynomial */ 
  iColor = 255 - 255.0 * ((double) n)/20; // nMax or lower walues in denominator
  return iColor;

 "if a 2-variable function z = f(x,y) has non-extremal critical points, i.e. it has saddle points, then it's best if the contour z heights are chosen so that the saddle points are on a contour, so that the crossing contours appear visually."Alan Ableson

How to choose parameters for which Level curves cross critical point ( and it's preimages ) ?

  • choose the parameter c such that it is on an escape line, then the critical value will be on an escape line as well
  • choose escape radius equal to n=th iteration of critical value
// find such ER for LSM/J that level curves croses critical point and it's preimages ( only for disconnected Julia sets)
double GiveER(int i_Max){

	complex double z= 0.0; // criical point
	int i;
	 ; // critical point escapes very fast here. Higher valus gives infinity
	for (i=0; i< i_Max; ++i ){
		z=z*z +c; 
	 return cabs(z);

Normalized iteration count (real escape time or fractional iteration or Smooth Iteration Count Algorithm (SICA))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/JEdit

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.[8]
    • 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[9]
/* 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 */ 

Density of level curves[10]

  "The spacing between level curves is a good way to estimate gradients: level curves that are close together represent areas of steeper descent/ascent." [11]

 "The density of the contour lines tells how steep is the slope of the terrain/function variation. When very close together it means f is varying rapidly (the elevation increase or decrease rapidly). When the curves are far from each other the variation is slower" [12]

Basin of attraction of finite attractor = interior of filled-in Julia setEdit

  • 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[13]
  • use edge detection for drawing Julia set

Internal Level SetsEdit

See :

  • algorithm 0 of program Mandel by Wolf Jung

How to choose size of attracting trap such that level curves cross at critical point ?

It depends on

  • dynamic type ( superattracting/attracting, parabolic, repelling)
  • period ( child period in the parabolic case)

  • petal in the parabolic case :
    • for period 1 and 2 : radius of a circle with parabolic point on it's boundary
    • for higher periods sector of the circle with center at parabolic point
  • for other cases ( except repelling) it is radius of the circle with attractor as a center

int local_setup(double cx){
	c = cx;
	zp = GiveFixed(c);
	switch ( DynamicType){
		case repelling: // no  interior = no attracting fixed point = only escaping points
		case attracting: 
			delta = sqrt(1.0 - 4.0* creal(c));  // delta is a distance between alfa and beta fixed points
			AR =  delta /20.0;
		case superattracting: // cabs(zp - zcr_last ) < PixelWidth 
			AR = 30.0* PixelWidth * iWidth / 5000 ; // 
		case parabolic:
				// zcr_last < parabolic_trap_center < zp
				int i; /* nr of point of critical orbit */
  				complex double z = zcr;
  				for (i=1;i<IterMax ; ++i) 
    					{ z = f(z); }
  				zcr_last = z;
				AR = (zp - zcr_last)/2.0;
				parabolic_trap_center = ( creal(zp) + creal(zcr_last))/ 2.0;
	AR2 = AR*AR;
	return 0;

// and print program info
fprintf (stdout, "DynamicType value is setup manually; Once can do it also numerically ( from multiplier of fixed point alfa or from some other properities)\n");
  		switch ( DynamicType){
		case repelling: 
				fprintf (stdout, "\tThere is only one Fatou basin: basin of infinity \n");
				fprintf (stdout, "\tthere is no interior = Julia set is disconnected \n");
				fprintf (stdout, "\tcritical point z=0 is repelling = attracted to infinity \n");
		case attracting: 
	  			fprintf (stdout, "\tbasin type is attracting \n");
	  			fprintf (stdout, "\tzcr_last =  %.16f \talfa fixed point zp = %.16f\n", creal (zcr_last), creal(zp));// 
	  			fprintf (stdout, "\tdelta =  %.16f is the distance between fixed points\n", delta);// 
	  			fprintf (stdout, "\tAtracting Radius AR is set manually  = %.16f = %f * PixelWidth = %f * ImageWidth \n", AR, AR / PixelWidth, AR /ImageWidth );
		case superattracting: 
				fprintf (stdout, "\tbasin type is superattracting \n");
	  			fprintf (stdout, "\tzcr =  %.16f  = zp = %.16f\n", creal (zcr), creal(zp));// 
	  			fprintf (stdout, "\tAtracting Radius AR is set manually  = %.16f = %f *PixelWidth = %f *ImageWidth \n", AR, AR / PixelWidth, AR /ImageWidth);
		case parabolic:
				fprintf (stdout, "\tbasin type  is parabolic \n");
				fprintf (stdout, "\tzcr_last =  %.16f < parabolic_trap_center = %.16f < zp = %.16f\n", creal (zcr_last), creal (parabolic_trap_center), creal(zp));// 
				fprintf (stdout, "\tzp - zcr_last =  %.16f AR*2 = %.16f \t difference = %.16f\n", creal (zp - zcr_last), AR *2.0, creal (zp - zcr_last) -  AR *2.0);// 
				fprintf (stdout, "\tAtracting Radius AR is tuned  = (zp - zcr_last)/2 = %.16f = %f *PixelWidth = %f *ImageWidth \n", AR, AR / PixelWidth, AR /ImageWidth);
				fprintf (stdout, "\tparabolic_trap_center z =  %.16f %+.16f*i  \n", creal (parabolic_trap_center), cimag (parabolic_trap_center));// parabolic_trap_center

Steps for parabolic basin

  • choose component with critical point inside
  • choose trap

Trap is a disk

  • inside component with critical point inside
  • trap has parabolic point on it's boundary
  • center of the trap is the midpoint between last point of critical orbit and fixed point
  • radius of the trap is half of distance between fixed point and last point of critical orbit

Decomposition of target setEdit


Binary decompositionEdit

Here color of pixel ( exterior of Julia set) is proportional to sign of imaginary part of last iteration ( cimag) = the radial borders of are at dyadic angles (shows external rays at dyadic angles).

Main loop is the same as in escape time.


  • Escape radius ( ER) should be bigger: ER = 200
  • attracting radius (AR)
    • for superattracting case small: AR = PixelWidth

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;


unsigned char ComputeColorOfBD (complex double z)

  double cabsz;

  int i;			// number of iteration
  for (i = 0; i < IterMax_LSM; ++i)

	cabsz = cabs(z); // numerical speed up : cabs(zp-z) = cabs(z) because zp = zcr = 0	

       if ( cabsz > ER  ||  cabsz < AR ) // if z is inside target set ( orbit trap) 
       			if (cimag(z) > 0) // binary decomposition of target set
       				{  return 0;}
       				else {return 255; }
      z = f(z);	


  return iColorOfUnknown;


Attracting case is used for "field lines" coloring method by Gertbuschmann

These curves :

  • are boundaries of binare decomposition boxes
  • are not field lines of potential = external rays
rotation of target set by internal angleEdit
 // for MBD
 double t0 = 1.0 / 3.0; // period = 3

// Modified BD
unsigned char ComputeColorOfMBD (complex double z)

  double cabsz;
  double turn; 

  int i;			// number of iteration
  for (i = 0; i < IterMax_LSM; ++i)

	cabsz = cabs(z); // numerical speed up : cabs2(zp-z) = cabs2(z) because zp = zcr = 0	

      //  if z is inside target set ( orbit trap) = exterior of circle with radius ER 
       if ( cabsz > ER  ) // exterior
       			if (creal(z) > 0) // binary decomposition of target set
       				{  return 0;}
       				else {return 255; }
      	if ( cabsz  < AR ) // if z is inside target set ( orbit trap) = interior of circle with radius AR
      			turn = c_turn(z);
      			if (turn < t0 || turn > t0+0.5) // modified binary decomposition of target set
      				{  return 0;}
       				else {return 255; }
      z = f(z);	


  return iColorOfUnknown;

Constant number of iterationsEdit

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:

  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 [16]

Inverse Iteration Method (IIM/J) : Julia setEdit

Inverse iteration of repellor for drawing Julia set

Complex potential - Boettcher coordinateEdit

See description here


This algorithm has 2 versions:

Compare it with version for parameter plane and Mandelbrot set : DEM/M It's the same as M-set exterior distance estimation but with derivative w.r.t. Z instead of w.r.t. C.


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

average discrete velocity of orbitEdit

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 BeckerEdit

"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 JungEdit

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 coordinateEdit

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 setEdit

Target set

  • trap for forward orbit
  • it is a set which captures any orbit tending to fixed / periodic point.

Julia setsEdit

"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 )[27]

  • 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 scanning [28]
    • edge detection

Fatou setEdit

Interior of filled Julia set can be coloured :

Periodic pointsEdit

More is here


One can make videos using :

  • zoom into dynamic plane
  • changing parametr c along path inside parameter plane[31]
  • changing coloring scheme ( for example color cycling )

Examples :

More tutorials and codeEdit


  1. Standard coloring algorithms from Ultra Fractal
  2. new fractalforum : lowest-optimal-bailout-values-for-the-mandelbrot-sets/
  3. math.stackexchange question: the-escape-radius-of-a-polynomial-and-its-filled-julia-set
  4. Faster Fractals Through Algebra by Bruce Dawson ( author of Fractal eXtreme)
  5. C code with gsl from tensorpudding
  6. Program Mandel by Wolf Jung on GNU General Public License
  7. Euler examples by R. Grothmann
  8. Drawing the Mandelbrot set by the method of escape lines. M. Romera et al.
  9. Julia Curves, Mandelbrot Set, Harold V. McIntosh.
  10. PythonDataScienceHandbook: density-and-contour-plots by Jake VanderPlas
  11. math.stackexchange question: what-do-level-curves-signify
  12. Contour lines by Rodolphe Vaillant
  13. The fixed points and periodic orbits by E Demidov
  14. Video : Julia Set Morphing with Magnetic Field lines by bryceguy72
  15. Video : Mophing Julia set with color bands / stripes by FreymanArt
  16. Gerard Westendorp : Platonic tilings of Riemann surfaces - 8 times iterated Automorphic function z->z^2 -0.1+ 0.75i
  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. Mark Braverman : On efficient computation of parabolic Julia sets
  29. Ray Tracing Quaternion Julia Sets on the GPU by Keenan Crane
  30. Tessellation of the Interior of Filled Julia Sets by Tomoki Kawahira
  31. Julia-Set-Animations at devianart