My CBSE Computer Science fractal project report. Radhika Prasad

1. 1
INTRODUCTION
Fractal definition: According to Mandelbrot (the one who coined the term
fractal), fractal is a rough or fragmented geometric shape that can be split
into parts, each of which is (at least approximately) reduced-sized copy of
the whole. They are made by repeating a process. Fractals are found
everywhere around us in nature. Ferns, rocky coastline, seashells, galaxies,
branching of blood vessels, lightning are a few examples.
My fascination for fractals was aroused when I got the book "Chaos" by
James Gleick. Fractal is an amazing topic which combines maths and art.
Beautiful patterns are generated from simple iterative functions. I decided
that for my class 12 project, I should try and recreate certain fractals using
my current C++ knowledge. In this project, I have made burning ship fractal,
Barnsley fern, Julia set, Mandelbrot set, Newton's fractal and Sierpinski
triangle. Each fractal is based on different algorithms. I have made
separate functions (by the name of the fractal) for each of the fractals. A
separate program was written to write information and history of each
fractal into a text file. Further, I have added an option were the user can
enter the zoom values and explore the fractals more. The parameters
entered to draw and zoom the fractals are stored in binary files. Along with
making the fractal images, I have made a function which tracks the path
followed by a point when it is successively iterated under the given function.
I have included the output when the entire program is run sequentially that
is all the fractals are displayed one after the other. Towards the end, the
user may insert, delete or modify certain parameters (these tasks are
implemented by file handling). All the parameters entered are then
displayed.
I enjoyed the challenge of writing the code and the beautiful outputs that I
got.

2. 2
Code
#include<fstream.h> //file handling
#include<conio.h>
#include<graphics.h>
#include<math.h> //sin, fabs, abs
#include<complex.h> //complex r(real, imaginary)
#include<stdlib.h> //rand()
#include<dos.h> //delay(milliseconds)
#include<stdio.h> //puts(string)
double pi=3.1415926536;
//NOTE: ALL THE FRACTAL IMAGES ARE REVERSED UPSIDE DOWN AS IN C++, y AXIS IS
POSITIVE DOWNWARDS
int w=639,h=479;
//julia
double cre_j=-0.7, cim_j=0.27015;
double newre_j,newim_j,oldre_j,oldim_j;
double zoom_j=1, X_j=0, Y_j=0;
int max_it_j=300;
//mandelbrot
double pre_m, pim_m;
double newre_m,newim_m,oldre_m,oldim_m;
double zoom_m=1, X_m=-0.5, Y_m=0;
int max_it_m=128;
//functions
void burning_ship();
void fern();
void juliadraw(int max_it_j, double zoom_j, double X_j, double Y_j, double
cre_j, double cim_j);
void mandelbrot_zoom(int max_it_m, double zoom_m, double X_m, double Y_m);
void mandelbrot_track();
void newton();
void sier_gask();
void filehandling_menu()
{
cout<<"n1. Insert parameters ";
cout<<"n2. Delete parameters ";
cout<<"n3. Modify parameters ";
cout<<"n4. Exit ";
}
class julia
{
public:
int max_it_j;
double X_j, Y_j, cre_j, cim_j, zoom_j;

3. 3
void getjuliainfo(char draw='y')
{
cout<<"Enter zoom value"; cin>>zoom_j;
cout<<"Enter x coordinate"; cin>>X_j;
cout<<"Enter y coordinate"; cin>>Y_j;
cout<<"Enter real part of c"; cin>>cre_j;
cout<<"enter imaginary part of c"; cin>>cim_j;
cout<<"enter max iterations"; cin>>max_it_j;
if(draw=='y')
{
juliadraw(max_it_j, zoom_j, X_j, Y_j, cre_j, cim_j);
}
}
void putjuliainfo()
{
cout<<"nZoom : "<<zoom_j;
cout<<"nx coordinate : "<<X_j;
cout<<"ny coordinate : "<<Y_j;
cout<<"nreal part of c : "<<cre_j;
cout<<"nimaginary part of c : "<<cim_j;
cout<<"nIterations : "<<max_it_j;
}
}Julia;
void displayjuliainfo()
{
cout<<"nJULIA PARAMETERS ENTERED BY YOUn";
ifstream f;
f.open("Julia.dat", ios::binary);
while(f.read((char*)&Julia, sizeof(Julia)))
{
Julia.putjuliainfo();
cout<<'n';
}
f.close();
}
void createjuliainfo()
{
juliadraw(max_it_j, zoom_j, X_j, Y_j, cre_j, cim_j);
ofstream f;
f.open("Julia.dat", ios::binary);
char choice='y';
cout<<"Do you want to zoom (y/n)? "; cin>>choice;
for(;choice!='n';)
{
Julia.getjuliainfo();
f.write((char*)&Julia, sizeof(Julia));
cout<<"Do you want to zoom (y/n) ";
cin>>choice;
}
f.close();
}
void Insertjuliaparameters()
{
ofstream f;
f.open("Julia.dat", ios::app|ios::nocreate);

4. 4
Julia.getjuliainfo('n');
f.write((char*)&Julia, sizeof(Julia));
f.close();
}
void Deletejuliaparameters()
{
int i=0, n;
cout<<"Which record do you want to delete (n value)? ";
cin>>n;
ifstream f;
ofstream ft;
f.open("Julia.dat", ios::binary);
ft.open("temp.dat", ios::binary);
while(f.read((char*)&Julia, sizeof(Julia)))
{
if(i==n-1)
{} //delete this record
else
ft.write((char*)&Julia, sizeof(Julia));
i++;
}
f.close();
ft.close();
remove("Julia.dat");
rename("temp.dat", "Julia.dat");
}
void Modifyjuliaparameters()
{
fstream f;
f.open("Julia.dat", ios::in|ios::out|ios::binary);
int n, i=0;
cout<<"Which record do you want to modify (n value)? ";
cin>>n;
long pos;
while(!f.eof())
{
pos=f.tellg(); //determine beginning pos of record
f.read((char*)&Julia, sizeof(Julia));
if(i==n-1) //this record is to be modified
{
Julia.getjuliainfo('n');
f.seekg(pos);
f.write((char*)&Julia, sizeof(Julia));
break;
}
if(f.eof()) break;
i++;
}
}

5. 5
class mandel
{
public:
int max_it_m;
double zoom_m, X_m, Y_m;
void getmandelinfo()
{
cout<<"Enter zoom value"; cin>>zoom_m;
cout<<"Enter x coordinate"; cin>>X_m;
cout<<"Enter y coordinate"; cin>>Y_m;
cout<<"enter max iterations"; cin>>max_it_m;
mandelbrot_zoom(max_it_m, zoom_m, X_m, Y_m);
}
void putmandelinfo()
{
cout<<"nZoom : "<<zoom_m;
cout<<"nx coordinate : "<<X_m;
cout<<"ny coordinate : "<<Y_m;
cout<<"nIterations : "<<max_it_m;
}
}Mandel;
void displaymandelinfo()
{
ifstream f;
f.open("Mandel.dat", ios::binary);
while(f.read((char*)&Mandel, sizeof(Mandel)))
{
Mandel.putmandelinfo();
cout<<'n';
}
f.close();
}
void createmandelinfo()
{
mandelbrot_zoom(max_it_m, zoom_m, X_m, Y_m);
ofstream f;
f.open("Mandel.dat", ios::binary);
char choice='y';
cout<<"Do you want to zoom (y/n)? "; cin>>choice;
for(;choice!='n';)
{
Mandel.getmandelinfo();
f.write((char*)&Mandel, sizeof(Mandel));
cout<<"Do you want to zoom (y/n) ";
cin>>choice;
}
f.close();
}
void brief_history()
{
char l[200];
int i=0;
ifstream f;

6. 6
f.open("fractal_history.txt");
while(f.getline(l,200))
{
if(i==23) break;
puts(l);
i++;
}
f.close();
}
void burn_theory()
{
char l[200];
ifstream f;
f.open("burnship_theory.txt");
while(f.getline(l,200))
{
puts(l);
}
f.close();
}
void fern_theory()
{
char l[200];
ifstream f;
f.open("fern_theory.txt");
while(f.getline(l,200))
{
puts(l);
}
f.close();
}
void julia_theory()
{
char l[200];
ifstream f;
f.open("julia_theory.txt");
while(f.getline(l,200))
{
puts(l);
}
f.close();
}
void mandelbrot_theory()
{
char l[200];
ifstream f;
f.open("mandel_theory.txt");
while(f.getline(l,200))
{
puts(l);
}
f.close();
}

7. 7
void newton_theory()
{
char l[200];
ifstream f;
f.open("newton_theory.txt");
while(f.getline(l,200))
{
puts(l);
}
f.close();
}
void sierpinski_theory()
{
char l[200];
ifstream f;
f.open("sier_theory.txt");
while(f.getline(l,200))
{
puts(l);
}
f.close();
}
void initialise() //initialise graphics
{
int gdriver = DETECT, gmode;
initgraph(&gdriver, &gmode, "C:turboc3bgi");
setcolor(1); //blue
}
void project_cover()
{
initialise();
setcolor(11); //light cyan
settextstyle(5,0,5);
outtextxy(20, 150, "Welcome to the world of fractals");
getch();
closegraph();
}
void main()
{
project_cover();
clrscr();
brief_history();
getch();
int choice_fractal;
clrscr();
cout<<"MENUnn";
cout<<"1. Sequential run";
cout<<"n2. Burning ship fractal";
cout<<"n3. Barnsley fern";

8. 8
cout<<"n4. Julia set";
cout<<"n5. Mandelbrot set";
cout<<"n6. Newton's fractal";
cout<<"n7. Sierpinski trianglen";
cin>>choice_fractal;
if(choice_fractal!=1)
{
switch(choice_fractal)
{
case 2:
clrscr();
burn_theory();
getch();
initialise();
burning_ship();
getch();
closegraph();
break;
case 3:
clrscr();
fern_theory();
getch();
initialise();
fern();
getch();
closegraph();
break;
case 4:
clrscr();
julia_theory();
getch();
initialise();
createjuliainfo();
getch();
closegraph();
break;
case 5:
clrscr();
mandelbrot_theory();
getch();
initialise();
createmandelinfo();
getch();
closegraph();
break;
case 6:
clrscr();
newton_theory();
getch();
initialise();
newton();
getch();
closegraph();
break;

9. 9
case 7:
clrscr();
sierpinski_theory();
getch();
initialise();
sier_gask();
getch();
closegraph();
break;
}
}
else
{
clrscr();
burn_theory();
getch();
initialise();
burning_ship();
getch();
closegraph();
clrscr();
fern_theory();
getch();
initialise();
fern();
getch();
closegraph();
clrscr();
julia_theory();
getch();
initialise();
createjuliainfo();
getch();
closegraph();
clrscr();
mandelbrot_theory();
getch();
initialise();
createmandelinfo();
getch();
closegraph();
initialise();
mandelbrot_track();
getch();
closegraph();
clrscr();
newton_theory();
getch();
initialise();
newton();

10. 10
getch();
closegraph();
clrscr();
sierpinski_theory();
getch();
initialise();
sier_gask();
getch();
closegraph();
}
clrscr();
int choice_file=0;
if((choice_fractal==1)||(choice_fractal==4)) //Julia
{
displayjuliainfo();
while(choice_file!=4)
{
getch();
cout<<endl;
filehandling_menu();
cout<<"nEnter your choice ";
cin>>choice_file;
switch(choice_file)
{
case 1: Insertjuliaparameters();
displayjuliainfo();
break;
case 2: Deletejuliaparameters();
displayjuliainfo();
break;
case 3: Modifyjuliaparameters();
displayjuliainfo();
break;
}
}
cout<<"nFinally ... ";
displayjuliainfo();
getch();
}
if((choice_fractal==1)||(choice_fractal==5)) //Mandelbrot
{
clrscr();
cout<<"MANDELBROT PARAMETERS ENTERED BY YOUn";
displaymandelinfo();
getch();
}
initialise();
setcolor(14); //Yellow
settextstyle(5, 0, 5); //settextstyle(int font,int dir, int size)
outtextxy(getmaxx()/2-150, getmaxy()/2-80, "THANK YOU");
getch();
}

11. 11
//burning ship
void burning_ship()
{
double pre, pim;
double newre,newim,oldre,oldim;
double zoom=0.7;
for(int x=0;x<w;x++)
for(int y=0;y<h;y++)
{
pre=1.5*(x-w/2)/(0.5*w*zoom);
pim=(y-h/2)/(0.5*h*zoom);
oldre=oldim=newre=newim=0;
int i;
for(i=0;i<300;i++)
{
oldre=newre;
oldim=newim;
newre=fabs(oldre)*fabs(oldre)-fabs(oldim)*fabs(oldim)+pre;
newim=2*fabs(oldre)*fabs(oldim)+pim; //fabs for float
if(newre*newre+newim*newim>4) //mod of the real number
should not exceed 2
{
putpixel(x,y,(i));
break;
}
}if(kbhit()){break;}
}
}
//fern
void fern()
{
int x,y;
double oldcart_x=0,newcart_x=0, oldcart_y=0,newcart_y=0;
int z=1;
for(long int i=0;i<500000;i++)
{
int r=rand()%100+1;
if(r==1)
{
newcart_x=0;
newcart_y=0.16*oldcart_y;
}
else if((r>1)&&(r<=86))
{
newcart_x=0.85*oldcart_x+0.04*oldcart_y;
newcart_y=-0.04*oldcart_x+0.85*oldcart_y+1.6;
}
else if((r>86)&&(r<=93))
{

12. 12
newcart_x=0.2*oldcart_x-0.26*oldcart_y;
newcart_y=0.23*oldcart_x+0.22*oldcart_y+1.6;
}
else
{
newcart_x=-0.15*oldcart_x+0.28*oldcart_y;
newcart_y=0.26*oldcart_x+0.24*oldcart_y+0.44;
}
if(kbhit()){break;}
oldcart_x=newcart_x;
oldcart_y=newcart_y;
putpixel((w*newcart_x+2.7*w)/(2.7*2*z), (h*newcart_y/10*z),2);
//putpixel(x,y,colour)
}
}
//julia
void juliadraw(int max_it_j, double zoom_j, double X_j, double Y_j, double
cre_j, double cim_j)
{ initialise();
{
for(int x=0;x<w;x++)
for(int y=0;y<h;y++)
{
newre_j=1.5*(x-w/2)/(0.5*w*zoom_j)+X_j;
newim_j=(y-h/2)/(0.5*h*zoom_j)+Y_j;
int i;
for(i=0;i<max_it_j;i++)
{
oldre_j=newre_j;
oldim_j=newim_j;
newre_j=oldre_j*oldre_j-oldim_j*oldim_j+cre_j;
newim_j=2*oldre_j*oldim_j+cim_j;
if(newre_j*newre_j+newim_j*newim_j>4)
{
putpixel(x,y,sqrt(i));
break;
}
}if(kbhit()){break;}
}
}
getch();
closegraph();
}
// mandelbrot zoom
void mandelbrot_zoom(int max_it_m, double zoom_m, double X_m, double Y_m)
{
initialise();
for(int x=0;x<w;x++)
for(int y=0;y<h;y++)
{
pre_m=1.5*(x-w/2)/(0.5*w*zoom_m)+X_m;

13. 13
pim_m=(y-h/2)/(0.5*h*zoom_m)+Y_m;
oldre_m=oldim_m=newre_m=newim_m=0;
int i;
for(i=0;i<max_it_m;i++)
{
oldre_m=newre_m;
oldim_m=newim_m;
newre_m=oldre_m*oldre_m-oldim_m*oldim_m+pre_m;
newim_m=2*oldre_m*oldim_m+pim_m;
if(newre_m*newre_m+newim_m*newim_m>4)
{
putpixel(x,y,sqrt(i));
break;
}
} if(kbhit()){break;}
}
getch();
closegraph();
}
// mandelbrot track a point
void mandelbrot_track()
{
double pre, pim;
double newre,newim,oldre,oldim;
int x=200,y=300;
pre=1.5*(x-w/2)/(0.5*w); //pre=-0.558685
pim=(y-h/2)/(0.5*h); //pim=0.254697
oldre=oldim=newre=newim=0;
int i;
cout<<"Tracking the journey of a point "<<"("<<pre<<", "<<pim<<")";
for(i=0;i<500;i++)
{
oldre=newre;
oldim=newim;
newre=oldre*oldre-oldim*oldim+pre;
newim=2*oldre*oldim+pim;
if(i==1)
setcolor(4); //all pts are red except 1st
outtextxy((w*newre+w*1.5)/3,(newim*h+h)/2,"*");
delay(200);
if(kbhit())
{break;}
}
}

14. 14
//newton
void newton()
{
double newre,newim;
complex r1(1.0,0.0);
complex r2(-0.5,sin(2*pi/3));
complex r3(-0.5,-sin(2*pi/3));
double conv=0.00001;
for(int x=0;x<w;x++)
for(int y=0;y<h;y++)
{
newre=(x-w/2)/(0.1*w);
newim=(y-h/2)/(0.1*w);
complex z(newre,newim);
int i;
for(i=0;i<300;i++)
{
while((abs(z-r1)>=conv)&&(abs(z-r2)>=conv)&&(abs(z-
r3)>=conv))
{
z=z-(z*z*z-1.0)/(3*z*z);
if(kbhit()){break;}
}
if(abs(z-r1)<conv)
{
putpixel(x,y,1); //blue
break;
}
if(abs(z-r2)<conv)
{
putpixel(x,y,2);
break;
}
if(abs(z-r3)<conv)
{
putpixel(x,y,4);
break;
}
if(kbhit())
{
break;
}
}if(kbhit()){break;}
}
}
//sierpinski gasket
void sier_gask()
{
line(10,10,510,10);
line(10,10,260,sqrt(3)*250);
line(510,10,260,sqrt(3)*250); //equilateral triangle

15. 15
int v1[2]={10,10};
int v2[2]={510,10};
int v3[3]={260,sqrt(3)*250}; //vertices
int vertex;
int newx,newy;
int x=rand()%600+10;
int y=rand()%400+10;
for(long int i=0;i<900000;i++)
{
vertex=rand()%3+1;
if(vertex==1)
{
newx=(x+v1[0])/2;
newy=(y+v1[1])/2;
putpixel(newx,newy,15);
}
if(vertex==2)
{
newx=(x+v2[0])/2;
newy=(y+v2[1])/2;
putpixel(newx,newy,15);
}
if(vertex==3)
{
newx=(x+v3[0])/2;
newy=(y+v3[1])/2;
putpixel(newx,newy,15);
}
x=newx;
y=newy;
}
}

16. 16
Separate program to write history and information
of each fractal into a text file
#include<fstream.h>
#include<conio.h>
#include<stdio.h>
void create()
{
ofstream f;
f.open("fractal_history.txt");
int i=0;
char l[200];
for(i=0;i<25;i++)
{
gets(l);
f<<l<<endl;
}
f.close();
}
void display()
{
char l[200];
ifstream f;
f.open("fractal_history.txt");
while(f.getline(l,200))
{
puts(l);
}
f.close();
}
void main()
{
clrscr();
create();
display();
getch();
}

17. 17
Program Output

18. 18

19. 19
1. Burning Ship

20. 20
2. Barnsley Fern

21. 21
3. Julia Set

22. 22
Julia set continued

23. 23
Julia set continued

24. 24
4. Mandelbrot Set

25. 25
Mandelbrot set continued

26. 26
Mandelbrot set continued

27. 27
5. Tracking a point in the Mandelbrot Set

28. 28
6. Newton’s Fractal

29. 29
7. Sierpinski Triangle

30. 30
8. File Handling

31. 31
File handling continued

32. 32
File handling continued

33. 33
File handling continued

34. 34
Some More Fractal Images
Mandelbrot Set

35. 35
Mandelbrot set images continued

36. 36
Mandelbrot set images continued

37. 37
Mandelbrot set images continued

38. 38
Mandelbrot set images continued

39. 39
Julia Set

40. 40
Julia set images continued

41. 41
Julia set images continued

42. 42
Julia set images continued

43. 43
Julia set images continued

44. 44
Julia set images continued

45. 45
Bibliography
http://en.wikipedia.org/wiki/Fractal
http://en.wikipedia.org/wiki/Burning_Ship_fractal
http://en.wikipedia.org/wiki/Barnsley_fern
http://en.wikipedia.org/wiki/Julia_set
http://en.wikipedia.org/wiki/Mandelbrot_set
http://en.wikipedia.org/wiki/Newton_fractal
http://en.wikipedia.org/wiki/Sierpinski_triangle
http://lodev.org/cgtutor/juliamandelbrot.html
http://www.mitchr.me/SS/newton/index.html
http://webserv.jcu.edu/math/vignettes/Julia.htm
https://prezi.com/v_njq-fu0ucd/fractals-for-dummies/
http://www.fractal.org/Bewustzijns-Besturings-
Model/Fractals-Useful-Beauty.htm
http://fractalfoundation.org/resources/what-are-fractals/