File:Julia set of the quadratic polynomial f(z) = z^2 - 1.12 + 0.222i.png
Original file (4,000 × 2,000 pixels, file size: 547 KB, MIME type: image/png)
Captions
Contents
Summary
[edit]DescriptionJulia set of the quadratic polynomial f(z) = z^2 - 1.12 + 0.222i.png |
English: This image shows the Julia set for a certain parameter in the quadratic family. It was created in response to a request by Professor Brandon Taylor (Ruskin School of Fine Art, Oxford and University of Southampton) for an image that would illustrate self-similarity of Julia sets.
The parameter c = -1.12 + 0.222i was chosen in the "dancing rabbits" component; i.e., the period 6 hyperbolic component bifurcating from the period 2 component of the Mandelbrot set. This choice was to create a distinctive subset of the Julia set that is clearly repeated at different scales. In addition, the choice was made to choose the multiplier to be non-real and with modulus close to 1, to create the distinctive "spiralling" structure near the repelling period two cycle, which again can be seen repeated throughout the Julia set. The (original) image was drawn using a general-purpose heuristic that estimates the distance to the Julia set using the size of the derivative and the distance to the beta-fixed point. See also: Étienne Ghys: Dynamics à la Dennis Sullivan[1] |
Date | |
Source | Own work |
Author | Lasse Rempe-Gillen |
Other versions |
|
Licensing
[edit]- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
Summary
[edit]- parameter c is inside period 6 components of Mandelbrot set with angled internal adress : .
- Input polynomial p(z)=(1+0i)*z^2+(-1.120000000000000+0.22200000000000000i)
- 1 critical points found cp#0: 0,0 . It's critical orbit is bounded and enters cycle #0 length=6
period 6 attracting orbit = limit cycle
[edit]- it's stability = |multiplier|=0.95429 =attractive
- cycle = { 0.23468426700349498404,-0.22442972485953274764 ; -1.1152919962215579464,0.11665974905508900983 ; 0.11026673978627110628,-0.038219368804712466892 ; -1.1093019662487375587,0.21357134961043047761 ; 0.064938130948894556838,-0.251830236114494177 ; -1.1792015069703463137,0.18929323030061190325 ; }
Period 2 repelling orbit
[edit]- z = -1.133085054887219 +0.175489847915920 i
- z = 0.133085054887219 -0.175489847915920 i
Orbit portrait of period 2 orbit with valence 3 is :[2]
Multiplier:
- radius = 1.00943e+00
- angle = 0.328869498392730553 (in turns)
m-describe by Claude Heiland-Allen
[edit]m-describe 200 6 10000 -1.12 0.222 8 the input point was -1.1200000000000000000000000000000000000000000000000000000000005e+00 + +2.2200000000000000000000000000000000000000000000000000000000002e-01 i the point didn't escape after 10000 iterations nearby hyperbolic components to the input point: - a period 1 cardioid with nucleus at +0e+00 + +0e+00 i the component has size 1.00000e+00 and is pointing west the atom domain has size 0.00000e+00 the atom domain coordinates of the input point are -nan + -nan i the atom domain coordinates in polar form are nan to the east the nucleus is 1.14179e+00 to the east of the input point the input point is exterior to this component at radius 1.36175e+00 and angle 0.477832873863374696 (in turns) the multiplier is -1.34856e+00 + +1.89052e-01 i a point in the attractor is -6.74286e-01 + +9.45244e-02 i external angles of this component are: .(0) .(1) - a period 2 circle with nucleus at -1e+00 + +0e+00 i the component has size 5.00000e-01 and is pointing west the atom domain has size 1.00000e+00 the atom domain coordinates of the input point are -0.12 + +0.222 i the atom domain coordinates in polar form are 0.25236 to the north-north-west the nucleus is 2.52357e-01 to the south-south-east of the input point the input point is exterior to this component at radius 1.00943e+00 and angle 0.328869498392730553 (in turns) the multiplier is -4.80000e-01 + +8.88000e-01 i a point in the attractor is +1.33045e-01 + -1.75343e-01 i - a period 6 circle with nucleus at -1.1380005e+00 + +2.4033237e-01 i the component has size 5.29939e-02 and is pointing north-north-west the atom domain has size 1.03599e-01 the atom domain coordinates of the input point are +0.18517 + +0.13088 i the atom domain coordinates in polar form are 0.22676 to the north-east the nucleus is 2.56924e-02 to the north-west of the input point the input point is interior to this component at radius 9.54286e-01 and angle 0.043426459533700348 (in turns) the multiplier is +9.18982e-01 + +2.57164e-01 i a point in the attractor is +1.1026669e-01 + -3.8219392e-02 i external angles of this component are: .(010110) .(011001)
c source code
[edit]/*
Adam Majewski
adammaj1 aaattt o2 dot pl // o like oxygen not 0 like zero
Structure of a program or how to analyze the program
============== Image X ========================
DrawImageOfX -> DrawPointOfX -> ComputeColorOfX
first 2 functions are identical for every X
check only last function = ComputeColorOfX
which computes color of one pixel !
==========================================
---------------------------------
indent d.c
default is gnu style
-------------------
c console progam
export OMP_DISPLAY_ENV="TRUE"
gcc d.c -lm -Wall -march=native -fopenmp
time ./a.out > b.txt
gcc d.c -lm -Wall -march=native -fopenmp
time ./a.out
time ./a.out >i.txt
time ./a.out >e.txt
convert -limit memory 1000mb -limit disk 1gb dd30010000_20_3_0.90.pgm -resize 2000x2000 10.png
*/
#include <stdio.h>
#include <stdlib.h> // malloc
#include <string.h> // strcat
#include <math.h> // M_PI; needs -lm also
#include <complex.h>
#include <omp.h> // OpenMP
#include <limits.h> // Maximum value for an unsigned long long int
// https://sourceforge.net/p/predef/wiki/Standards/
#if defined(__STDC__)
#define PREDEF_STANDARD_C_1989
#if defined(__STDC_VERSION__)
#if (__STDC_VERSION__ >= 199409L)
#define PREDEF_STANDARD_C_1994
#endif
#if (__STDC_VERSION__ >= 199901L)
#define PREDEF_STANDARD_C_1999
#endif
#endif
#endif
/* --------------------------------- global variables and consts ------------------------------------------------------------ */
// virtual 2D array and integer ( screen) coordinate
// Indexes of array starts from 0 not 1
//unsigned int ix, iy; // var
static unsigned int ixMin = 0; // Indexes of array starts from 0 not 1
static unsigned int ixMax; //
static unsigned int iWidth; // horizontal dimension of array
static unsigned int iyMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iyMax; //
static unsigned int iHeight = 10000; //
// The size of array has to be a positive constant integer
static unsigned long long int iSize; // = iWidth*iHeight;
// memmory 1D array
unsigned char *data;
unsigned char *edge;
//unsigned char *edge2;
// unsigned int i; // var = index of 1D array
//static unsigned int iMin = 0; // Indexes of array starts from 0 not 1
static unsigned int iMax; // = i2Dsize-1 =
// The size of array has to be a positive constant integer
// unsigned int i1Dsize ; // = i2Dsize = (iMax -iMin + 1) = ; 1D array with the same size as 2D array
// see SetPlane
double radius = 0.9;
complex double center = 0.0;
double DisplayAspectRatio = 2.0; // https://en.wikipedia.org/wiki/Aspect_ratio_(image)
// dx = dy compare setup : iWidth = iHeight;
double ZxMin; //= -1.3; //-0.05;
double ZxMax;// = 1.3; //0.75;
double ZyMin;// = -1.3; //-0.1;
double ZyMax;// = 1.3; //0.7;
double PixelWidth; // =(ZxMax-ZxMin)/ixMax;
double PixelHeight; // =(ZyMax-ZyMin)/iyMax;
double ratio;
/*
ER = pow(10,ERe);
AR = pow(10,-ARe);
*/
//int ARe ; // increase ARe until black ( unknown) points disapear
//int ERe ;
double ER;
double ER2; //= 1e60;
double AR; // bigger values do not works
double AR2;
double AR12;
int IterMax = 100000;
/* colors = shades of gray from 0 to 255
unsigned char colorArray[2][2]={{255,231}, {123,99}};
color = 245; exterior
*/
unsigned char iColorOfExterior = 245;
unsigned char iColorOfInterior1 = 99;
unsigned char iColorOfInterior2 = 183;
unsigned char iColorOfBoundary = 0;
unsigned char iColorOfUnknown = 5;
// pixel counters
unsigned long long int uUnknown = 0;
unsigned long long int uInterior = 0;
unsigned long long int uExterior = 0;
// periodic points = attractors
int iPeriod = 6;
complex double zp6= 0.11026673978627110628-0.038219368804712466892*I ; // period 6
/*
f(z)=z^6+A*z+c
A=(-33725751,810162*i)*2^-25
c=(-3096576+8798208*i)*2^-25
c is case sensitive
changed to lower because A is used
*/
char * sFunction = "f(z) = z^2 +c where c = -1.12 + 0.222*I";
complex double c = -1.12 + 0.222*I;
/* ------------------------------------------ functions -------------------------------------------------------------*/
//------------------complex numbers -----------------------------------------------------
// from screen to world coordinate ; linear mapping
// uses global cons
double
GiveZx (int ix)
{
return (ZxMin + ix * PixelWidth);
}
// uses globaal cons
double
GiveZy (int iy)
{
return (ZyMax - iy * PixelHeight);
} // reverse y axis
complex double
GiveZ (int ix, int iy)
{
double Zx = GiveZx (ix);
double Zy = GiveZy (iy);
return Zx + Zy * I;
}
double cabs2(complex double z){
return creal(z)*creal(z)+cimag(z)*cimag(z);
}
//A=(-33725751,810162*i)*2^-25
//c=(-3096576+8798208*i)*2^-25
complex double ToComplexDouble( double m, double n){
return (m+n*I)/pow(2.0,25.0);
}
// =====================
int IsPointInsideTrap1(complex double z){
if ( cabs2(z - zp6) < AR2) {return 1;} // circle with prabolic point zp on it's boundary
return 0; // outside
}
// =====================
int IsPointInsideTrap2(complex double z){
//if (cabs2(z - zp2) <AR2) {return 1;} // circle around periodic point
return 0; // outside
}
// ****************** DYNAMICS = trap tests ( target sets) ****************************
/* ----------- array functions = drawing -------------- */
/* gives position of 2D point (ix,iy) in 1D array ; uses also global variable iWidth */
unsigned int
Give_i (unsigned int ix, unsigned int iy)
{
return ix + iy * iWidth;
}
// f(z)=1+z−3z2−3.75z3+1.5z4+2.25z5
unsigned char
ComputeColor_Fatou (complex double z, int IterMax)
{
double r2;
int i; // number of iteration
for (i = 0; i < IterMax; ++i)
{
z = z*z +c; // complex iteration f(z)=z^6+A*z+c
r2 =cabs2(z);
if (r2 > ER2) // esaping = exterior
{
uExterior += 1;
return iColorOfExterior;
}
if ( IsPointInsideTrap1(z)) {
uInterior +=1;
return 100 + 20*(i % iPeriod); }
}
uUnknown += 1;
return iColorOfUnknown;
}
// plots raster point (ix,iy)
int
DrawFatouPoint (unsigned char A[], int ix, int iy, int IterMax)
{
int i; /* index of 1D array */
unsigned char iColor = 0;
complex double z;
i = Give_i (ix, iy); /* compute index of 1D array from indices of 2D array */
z = GiveZ (ix, iy);
iColor = ComputeColor_Fatou (z, IterMax);
A[i] = iColor; // interior
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int
DrawFatouImage (unsigned char A[], int IterMax)
{
unsigned int ix, iy; // pixel coordinate
fprintf (stdout, "compute Fatou image \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, uUnknown, uInterior, uExterior)
for (iy = iyMin; iy <= iyMax; ++iy)
{
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix)
DrawFatouPoint (A, ix, iy, IterMax); //
}
return 0;
}
//=========
int IsInside (int x, int y, int xcenter, int ycenter, int r){
double dx = x- xcenter;
double dy = y - ycenter;
double d = sqrt(dx*dx+dy*dy);
if (d<r)
return 1;
return 0;
}
int PlotBigPoint(complex double z, unsigned char A[]){
unsigned int ix_seed = (creal(z)-ZxMin)/PixelWidth;
unsigned int iy_seed = (ZyMax - cimag(z))/PixelHeight;
unsigned int i;
/* mark seed point by big pixel */
int iSide =3.0*iWidth/2000.0 ; /* half of width or height of big pixel */
int iY;
int iX;
for(iY=iy_seed-iSide;iY<=iy_seed+iSide;++iY){
for(iX=ix_seed-iSide;iX<=ix_seed+iSide;++iX){
if (IsInside(iX, iY, ix_seed, iy_seed, iSide)) {
i= Give_i(iX,iY); /* index of _data array */
A[i]= 255-A[i];}}}
return 0;
}
// fill array
// uses global var : ...
// scanning complex plane
int MarkAttractors (unsigned char A[])
{
fprintf (stderr, "mark attractors \n");
PlotBigPoint(zp6, A); // period 6 attracting cycle
return 0;
}
// =====================
int IsPointInsideTraps(unsigned int ix, unsigned int iy){
complex double z = GiveZ (ix, iy);
if ( IsPointInsideTrap1(z)) {return 1;} // circle with prabolic point on it's boundary
//if (IsPointInsideTrap2(z)) {return 1;}
return 0; // outside
}
int MarkTraps(unsigned char A[]){
unsigned int ix, iy; // pixel coordinate
unsigned int i;
fprintf (stderr, "Mark traps \n");
// for all pixels of image
#pragma omp parallel for schedule(dynamic) private(ix,iy) shared(A, ixMax , iyMax, uUnknown, uInterior, uExterior)
for (iy = iyMin; iy <= iyMax; ++iy)
{
fprintf (stderr, " %d from %d \r", iy, iyMax); //info
for (ix = ixMin; ix <= ixMax; ++ix){
if (IsPointInsideTraps(ix, iy)) {
i= Give_i(ix,iy); /* index of _data array */
A[i]= 255-A[i]; // inverse color
}}}
return 0;
}
int PlotPoint(complex double z, unsigned char A[]){
unsigned int ix = (creal(z)-ZxMin)/PixelWidth;
unsigned int iy = (ZyMax - cimag(z))/PixelHeight;
unsigned int i = Give_i(ix,iy); /* index of _data array */
A[i]= 255-A[i]; // Mark point with inveres color
return 0;
}
// ***********************************************************************************************
// ********************** edge detection usung Sobel filter ***************************************
// ***************************************************************************************************
// from Source to Destination
int ComputeBoundaries(unsigned char S[], unsigned char D[])
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
/* sobel filter */
unsigned char G, Gh, Gv;
// boundaries are in D array ( global var )
// clear D array
memset(D, iColorOfExterior, iSize*sizeof(*D)); // for heap-allocated arrays, where N is the number of elements = FillArrayWithColor(D , iColorOfExterior);
// printf(" find boundaries in S array using Sobel filter\n");
#pragma omp parallel for schedule(dynamic) private(i,iY,iX,Gv,Gh,G) shared(iyMax,ixMax)
for(iY=1;iY<iyMax-1;++iY){
for(iX=1;iX<ixMax-1;++iX){
Gv= S[Give_i(iX-1,iY+1)] + 2*S[Give_i(iX,iY+1)] + S[Give_i(iX-1,iY+1)] - S[Give_i(iX-1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX+1,iY-1)];
Gh= S[Give_i(iX+1,iY+1)] + 2*S[Give_i(iX+1,iY)] + S[Give_i(iX-1,iY-1)] - S[Give_i(iX+1,iY-1)] - 2*S[Give_i(iX-1,iY)] - S[Give_i(iX-1,iY-1)];
G = sqrt(Gh*Gh + Gv*Gv);
i= Give_i(iX,iY); /* compute index of 1D array from indices of 2D array */
if (G==0) {D[i]=255;} /* background */
else {D[i]=0;} /* boundary */
}
}
return 0;
}
// copy from Source to Destination
int CopyBoundaries(unsigned char S[], unsigned char D[])
{
unsigned int iX,iY; /* indices of 2D virtual array (image) = integer coordinate */
unsigned int i; /* index of 1D array */
//printf("copy boundaries from S array to D array \n");
for(iY=1;iY<iyMax-1;++iY)
for(iX=1;iX<ixMax-1;++iX)
{i= Give_i(iX,iY); if (S[i]==0) D[i]=0;}
return 0;
}
// *******************************************************************************************
// ********************************** save A array to pgm file ****************************
// *********************************************************************************************
int
SaveArray2PGMFile (unsigned char A[], int a, int b, int c, char *comment)
{
FILE *fp;
const unsigned int MaxColorComponentValue = 255; /* color component is coded from 0 to 255 ; it is 8 bit color file */
char name[100]; /* name of file */
snprintf (name, sizeof name, "%d_%d_%d", a, b, c ); /* */
char *filename = strcat (name, ".pgm");
char long_comment[200];
sprintf (long_comment, "%s %s", sFunction, comment);
// save image array to the pgm file
fp = fopen (filename, "wb"); // create new file,give it a name and open it in binary mode
fprintf (fp, "P5\n # %s\n %u %u\n %u\n", long_comment, iWidth, iHeight, MaxColorComponentValue); // write header to the file
fwrite (A, iSize, 1, fp); // write array with image data bytes to the file in one step
fclose (fp);
// info
printf ("File %s saved ", filename);
if (long_comment == NULL || strlen (long_comment) == 0)
printf ("\n");
else
printf (". Comment = %s \n", long_comment);
return 0;
}
int
PrintCInfo ()
{
printf ("gcc version: %d.%d.%d\n", __GNUC__, __GNUC_MINOR__, __GNUC_PATCHLEVEL__); // https://stackoverflow.com/questions/20389193/how-do-i-check-my-gcc-c-compiler-version-for-my-eclipse
// OpenMP version is displayed in the console : export OMP_DISPLAY_ENV="TRUE"
printf ("__STDC__ = %d\n", __STDC__);
printf ("__STDC_VERSION__ = %ld\n", __STDC_VERSION__);
printf ("c dialect = ");
switch (__STDC_VERSION__)
{ // the format YYYYMM
case 199409L:
printf ("C94\n");
break;
case 199901L:
printf ("C99\n");
break;
case 201112L:
printf ("C11\n");
break;
case 201710L:
printf ("C18\n");
break;
//default : /* Optional */
}
return 0;
}
int
PrintProgramInfo ()
{
// display info messages
printf ("Numerical approximation of Julia set %s\n", sFunction);
//printf ("iPeriodParent = %d \n", iPeriodParent);
//printf ("iPeriodOfChild = %d \n", iPeriodChild);
//printf ("parameter A = ( %.16f ; %.16f ) \n", creal (a), cimag (a));
printf ("parameter c = ( %.16f ; %.16f ) \n", creal (c), cimag (c));
printf ("Image Width = %f in world coordinate\n", ZxMax - ZxMin);
printf ("PixelWidth = %.16f \n", PixelWidth);
printf ("AR = %.16f = %f *PixelWidth\n", AR, AR / PixelWidth);
printf("pixel counters\n");
printf ("uUnknown = %llu\n", uUnknown);
printf ("uExterior = %llu\n", uExterior);
printf ("uInterior = %llu\n", uInterior);
printf ("Sum of pixels = %llu\n", uInterior+uExterior + uUnknown);
printf ("all pixels of the array = iSize = %llu\n", iSize);
// image corners in world coordinate
// center and radius
// center and zoom
// GradientRepetition
printf ("Maximal number of iterations = iterMax = %d \n", IterMax);
printf ("ratio of image = %f ; it should be 1.000 ...\n", ratio);
//
return 0;
}
int SetPlane(complex double center, double radius, double a_ratio){
ZxMin = creal(center) - radius*a_ratio;
ZxMax = creal(center) + radius*a_ratio; //0.75;
ZyMin = cimag(center) - radius; // inv
ZyMax = cimag(center) + radius; //0.7;
return 0;
}
// *****************************************************************************
//;;;;;;;;;;;;;;;;;;;;;; setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
// **************************************************************************************
int
setup ()
{
fprintf (stderr, "setup start\n");
/* 2D array ranges */
iWidth = iHeight* DisplayAspectRatio ;
iSize = iWidth * iHeight; // size = number of points in array
// iy
iyMax = iHeight - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
//ix
ixMax = iWidth - 1;
/* 1D array ranges */
// i1Dsize = i2Dsize; // 1D array with the same size as 2D array
iMax = iSize - 1; // Indexes of array starts from 0 not 1 so the highest elements of an array is = array_name[size-1].
SetPlane( center, radius, DisplayAspectRatio );
/* Pixel sizes */
PixelWidth = (ZxMax - ZxMin) / ixMax; // ixMax = (iWidth-1) step between pixels in world coordinate
PixelHeight = (ZyMax - ZyMin) / iyMax;
ratio = ((ZxMax - ZxMin) / (ZyMax - ZyMin)) / ((double) iWidth / (double) iHeight); // it should be 1.000 ...
ER = 2.0; //
ER2 = ER*ER;
AR = PixelWidth*20.0*iWidth/2000.0 ; //
AR2 = AR * AR;
AR12 = AR/2.0;
//sFunction="";
//sprintf (sFunction, "%f %+f", creal(c), cimag(c) ); // add c
/* create dynamic 1D arrays for colors ( shades of gray ) */
data = malloc (iSize * sizeof (unsigned char));
edge = malloc (iSize * sizeof (unsigned char));
if (data == NULL || edge == NULL)
{
fprintf (stderr, " Could not allocate memory");
return 1;
}
fprintf (stderr, " end of setup \n");
return 0;
} // ;;;;;;;;;;;;;;;;;;;;;;;;; end of the setup ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
int
end ()
{
fprintf (stderr, " allways free memory (deallocate ) to avoid memory leaks \n"); // https://en.wikipedia.org/wiki/C_dynamic_memory_allocation
free (data);
free(edge);
PrintProgramInfo ();
PrintCInfo ();
return 0;
}
// ********************************************************************************************************************
/* ----------------------------------------- main -------------------------------------------------------------*/
// ********************************************************************************************************************
int
main ()
{
setup ();
DrawFatouImage (data, IterMax); // first find Fatou
SaveArray2PGMFile (data, iWidth, IterMax, 0, "Fatou, name = iWidth_IterMax_n");
ComputeBoundaries(data,edge);
SaveArray2PGMFile (edge, iWidth, IterMax, 1, "Boundaries of Fatou; name = iWidth_IterMax_n");
CopyBoundaries(edge,data);
SaveArray2PGMFile (data, iWidth, IterMax, 2, "Fatou with boundaries; name = iWidth_IterMax_n");
//MarkAttractors(data);
MarkTraps(data);
SaveArray2PGMFile (data, iWidth, IterMax, 4, "Fatou with boundaries and traps; name = iWidth_IterMax_n");
end ();
return 0;
}
text output
[edit]OPENMP DISPLAY ENVIRONMENT BEGIN _OPENMP = '201511' OMP_DYNAMIC = 'FALSE' OMP_NESTED = 'FALSE' OMP_NUM_THREADS = '8' OMP_SCHEDULE = 'DYNAMIC' OMP_PROC_BIND = 'FALSE' OMP_PLACES = '' OMP_STACKSIZE = '0' OMP_WAIT_POLICY = 'PASSIVE' OMP_THREAD_LIMIT = '4294967295' OMP_MAX_ACTIVE_LEVELS = '2147483647' OMP_CANCELLATION = 'FALSE' OMP_DEFAULT_DEVICE = '0' OMP_MAX_TASK_PRIORITY = '0' OMP_DISPLAY_AFFINITY = 'FALSE' OMP_AFFINITY_FORMAT = 'level %L thread %i affinity %A' OPENMP DISPLAY ENVIRONMENT END setup start end of setup compute Fatou image File 20000_100000_0.pgm saved . Comment = f(z) = z^2 +c where c = -1.12 + 0.222*I Fatou, name = iWidth_IterMax_n File 20000_100000_1.pgm saved . Comment = f(z) = z^2 +c where c = -1.12 + 0.222*I Boundaries of Fatou; name = iWidth_IterMax_n File 20000_100000_2.pgm saved . Comment = f(z) = z^2 +c where c = -1.12 + 0.222*I Fatou with boundaries; name = iWidth_IterMax_n File 20000_100000_4.pgm saved . Comment = f(z) = z^2 +c where c = -1.12 + 0.222*I Fatou with boundaries and traps; name = iWidth_IterMax_n Numerical approximation of Julia set f(z) = z^2 +c where c = -1.12 + 0.222*I parameter c = ( -1.1200000000000001 ; 0.2220000000000000 ) Image Width = 3.600000 in world coordinate PixelWidth = 0.0001800090004500 AR = 0.0360018000900045 = 200.000000 *PixelWidth pixel counters uUnknown = 0 uExterior = 91185244 uInterior = 18477474 Sum of pixels = 109662718 all pixels of the array = iSize = 200000000 Maximal number of iterations = iterMax = 100000 ratio of image = 1.000000 ; it should be 1.000 ... gcc version: 9.3.0 __STDC__ = 1 __STDC_VERSION__ = 201710 c dialect = C18 Mark traps 9999 allways free memory (deallocate ) to avoid memory leaks real 0m13,670s user 1m26,360s sys 0m1,039s
Image Magic src code
[edit]convert 20000_100000_2.pgm -resize 4000x2000 2_4.png
References
[edit]- ↑ Étienne Ghys: Dynamics à la Dennis Sullivan
- ↑ John W Milnor : Periodic Orbits, Externals Rays and the Mandelbrot Set: An Expository Account ,1999 , arxiv=math/9905169
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current | 17:41, 15 August 2020 | 4,000 × 2,000 (547 KB) | Soul windsurfer (talk | contribs) | better quality | |
19:21, 30 May 2014 | 3,800 × 1,560 (369 KB) | JeffyP (talk | contribs) | I'm here to make the world a smaller place ☺ | ||
09:44, 7 March 2013 | 3,800 × 1,560 (1.12 MB) | L rempe (talk | contribs) | User created page with UploadWizard |
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File change date and time | 21:11, 15 August 2020 |