The graph is a set of complex numbers. You take each number to examine (corresponding to a pixel) and perform a simple equation a number of times to see if it grows towards infinity (which you could colour white) otherwise colour it black:

More colourful fractals are used where the colour indicates how quickly the result grows towards infinity.

The equation used is:

z

Set z

It is a bit like if you take a number and repeatedly square it. If you keep squaring 2, this happens:

2, 4, 16, 256, 65536, ...

As you can see this grows towards infinity so colour it white.

Le'ts do it to 0.5:

0.5, 0.25, 0.0625, ...

This grows towards 0 so colour it black.

Let's do it to the number 1

1, 1, 1, 1, ...

Colour this black also

ViewWindow -2,1.5,.5,-1.5,1.5,.5

For Ymin→T To Ymax Step (Ymax-Ymin)÷62

For Xmin→S To Xmax Step (Xmax-Xmin)÷125

S+Ti→C~D

1→I

Do

C^{2}+D→C

1+I→I

LpWhile Abs C≤2 And I<20

Abs C≤2⇒Plot S,T

Next

Next

For Ymin→T To Ymax Step (Ymax-Ymin)÷62

For Xmin→S To Xmax Step (Xmax-Xmin)÷125

S+Ti→C~D

1→I

Do

C

1+I→I

LpWhile Abs C≤2 And I<20

Abs C≤2⇒Plot S,T

Next

Next

Here is a code fragment written in Java (easily converted to C++)

class MandelbrotPanel extends JPanel

{

public static final int MAX_ITERATIONS=256;

//maximum 256 for a palette from 0 to 255

public void paintComponent(Graphics g)

{

//Range of the graph

float GraphTop=1.5F,GraphBottom=-1.5F,GraphLeft=-2F,GraphRight=1.5F;

int x,y,iterations; //x and y are the current pixel to draw

Rectangle r = getBounds();//this is just to get the applet dimensions

float IncrementX=((GraphRight-GraphLeft)/(r.width-1));//this is the increment on the graph corresponding to one pixel

float DecrementY=((GraphTop-GraphBottom)/(r.height-1));

float Zx,Zy,CoordReal,CoordImaginary=GraphTop,SquaredX,SquaredY;

int palette[] = new int[256];

for(int n=0;n<256;n++)

{

//I am making a palette which is non-linear to make it brighter

palette[n]=(int)(n+512-512*Math.exp(-n/50.0)/3.0);

palette[n]=palette[n]<<16 | palette[n]<<8 | palette[n];

/*

* The maximum value should be 255.??? so the (int) conversion

* should give 255 as the maximum value

*/

}

palette[255]=0;

//The last colour index is for values clearly within the Mandelbrot set, coloured black

/*

CoordReal and CoordImaginary are the variables of the point on the graph. Remember that (0,0) on the screen correspond to GraphLeft,GraphTop.

SquaredX and SquaredY are used just to speed up the program

*/

for(y=0;y<r.height;y++)

{

CoordReal=GraphLeft;//reset the variable when you start a new line

for(x=0;x<r.width;x++)

{

iterations=0;

Zx=CoordReal;Zy=CoordImaginary;//prepare to calculate z^2 + c

SquaredX=Zx*Zx;SquaredY=Zy*Zy; //set the initial squares

do

{

Zy=Zx*Zy;

Zy=Zy+Zy+CoordImaginary;//adding Zy to itself removes a slower times two multiply operation

Zx=SquaredX-SquaredY+CoordReal;

SquaredX=Zx*Zx;SquaredY=Zy*Zy;

iterations++;

}while((iterations<MAX_ITERATIONS)&&((SquaredX+SquaredY)<4.0));

//Squareroot(n) < 2 squaring both sides gives n < 4

iterations--;//iterations would've otherwise been from 1 to MAX_ITERATIONS

g.setColor(new Color(palette[iterations]));

g.drawLine(x,y,x,y);//This plots one pixel, use a different command

CoordReal+=IncrementX;//Increment to the next place on the graph

}

CoordImaginary-=DecrementY;//Go down one line on the graph

}

}

}

{

public static final int MAX_ITERATIONS=256;

//maximum 256 for a palette from 0 to 255

public void paintComponent(Graphics g)

{

//Range of the graph

float GraphTop=1.5F,GraphBottom=-1.5F,GraphLeft=-2F,GraphRight=1.5F;

int x,y,iterations; //x and y are the current pixel to draw

Rectangle r = getBounds();//this is just to get the applet dimensions

float IncrementX=((GraphRight-GraphLeft)/(r.width-1));//this is the increment on the graph corresponding to one pixel

float DecrementY=((GraphTop-GraphBottom)/(r.height-1));

float Zx,Zy,CoordReal,CoordImaginary=GraphTop,SquaredX,SquaredY;

int palette[] = new int[256];

for(int n=0;n<256;n++)

{

//I am making a palette which is non-linear to make it brighter

palette[n]=(int)(n+512-512*Math.exp(-n/50.0)/3.0);

palette[n]=palette[n]<<16 | palette[n]<<8 | palette[n];

/*

* The maximum value should be 255.??? so the (int) conversion

* should give 255 as the maximum value

*/

}

palette[255]=0;

//The last colour index is for values clearly within the Mandelbrot set, coloured black

/*

CoordReal and CoordImaginary are the variables of the point on the graph. Remember that (0,0) on the screen correspond to GraphLeft,GraphTop.

SquaredX and SquaredY are used just to speed up the program

*/

for(y=0;y<r.height;y++)

{

CoordReal=GraphLeft;//reset the variable when you start a new line

for(x=0;x<r.width;x++)

{

iterations=0;

Zx=CoordReal;Zy=CoordImaginary;//prepare to calculate z^2 + c

SquaredX=Zx*Zx;SquaredY=Zy*Zy; //set the initial squares

do

{

Zy=Zx*Zy;

Zy=Zy+Zy+CoordImaginary;//adding Zy to itself removes a slower times two multiply operation

Zx=SquaredX-SquaredY+CoordReal;

SquaredX=Zx*Zx;SquaredY=Zy*Zy;

iterations++;

}while((iterations<MAX_ITERATIONS)&&((SquaredX+SquaredY)<4.0));

//Squareroot(n) < 2 squaring both sides gives n < 4

iterations--;//iterations would've otherwise been from 1 to MAX_ITERATIONS

g.setColor(new Color(palette[iterations]));

g.drawLine(x,y,x,y);//This plots one pixel, use a different command

CoordReal+=IncrementX;//Increment to the next place on the graph

}

CoordImaginary-=DecrementY;//Go down one line on the graph

}

}

}

I have made a non-linear palette just to brighten it up. It is made just for 256 iterations only so if you lessen the iterations, you should change this.

I made this to iterate a maximum of 256 times but you can do one that iterates 50 times with perfectly good results and of course it will be faster.

I have set a variable to Zx*Zy just so I do not have to multiply these again in the loop.

Now, to square a complex number, I expand this equation:

(Zx + Zyi)

Zx × Zx + Zx × Zy +Zx × Zy - Zy×Zy =

Zx

The real part is Zx

The imaginary part is 2(Zx×Zy). It is quicker to set a variable n = Zx*Zy then set n = n + n to avoid multiplying by two (adding is quicker than multiplying). Zy is a floating point number so I cannot do a bit shift left to multiply by two.

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