The Julia Set and Mandelbrot Set are those quite well known sets on the complex
plane that create those pretty infinitely detailed images. They're so pretty,
that there is even art created with them. The Mandelbrot Set isn't a real
fractals by definition, but it's semi self similar and still shows infinite
detail, so it's usually called a fractal as well.

Research on Julia Sets was done in 1917 by Gaston Julia himself, but he didn't
have a computer available to actually draw it. The topic didn't gain much
interest, until in the 1970s, Benout Mandelbrot drew Julia Sets on a computer,
and discovered the Mandelbrot set.

This tutorial will only cover the basics of the theory (there's much more to
tell about the Julia and Mandelbrot sets), and the code to draw them. Apart from
that, more code is included of some pretty handy fractal viewers where you can
zoom and move around on the fly.

Julia Set

This chapter first tries to explain a bit of the mathematical formulas behind
Julia Sets, more specifically, Quadratic Julia Sets. Knowledge about complex
numbers is required, so if you need it you can read the appendix about complex
numbers.

So how to generate such a beautiful fractal? In short:
for every pixel, iterate znew = zold + c on
the complex plane until it leaves the circle around the origin with radius 2.
The number of iterations it the color of the pixel.

The screen will be representing a part of the complex plane, inside the
circle with radius 2 around the origin. For a pixel, the x coordinate will
represent the real part of it's complex coordinates, and the y coordinate will
be the imaginary part.

For a julia set, for each pixel apply an iterated complex function. This
function is newz = oldz + c, with z and c both being complex numbers. Z is
initially the coordinates of the pixel, and will then constantly be updated
through every iteration: each iteration, the "newz" of the previous iteration is
now used as "oldz".

If you keep iterating this function, depending on the initial condition (the
pixel), z will either go to infinity, or remain in the circle with radius 2
around the origin of the complex plane forever. The points that remain in the
circle forever, are the ones that belong to the Julia Set. So keep iterating the
function until the distance of z to the origin (0,0) is greater than 2. Also
give a maximum number of iterations, for example 256, or the computer would be
stuck in an endless loop.

The color value of the pixel will then become the number of times we had to
iterate the function before the distance of z to the origin got larger than 2.
The constant c in the formula can be anything really, as long as it's also
inside the circle with radius 2. Different values of c give different Julia
Sets. Some Julia Sets are connected, others aren't. The Mandelbrot Set is the
collection of all points c that generate a connected Julia Set.

Here's an example of the calculations:

First, you can choose a constant c for the function, which one you choose
will determinate the shape of the fractal. Let's take c = (-0.5,0.5) in this
example, so -0.5 is the real part and 0.5 the imaginary part.

Imagine we're currently calculating the color of pixel (256,192) on a 256*256
screen. First, we transform the coordinates so it lies between -1 and 1 (if you
zoom or move around in the fractal a different transformation is required): the
coordinates become (1,0.5) so p = 1 + 0.5i.

The distance is now 8.78515625, so we're outside the circle with radius 2, and
we now know the point is outside the julia set. The number of iterations was now
only 2, so the pixel gets color value 2 and we're done. Some start values give
more than 256 iterations, and depending on how much maximal iterations you've
set, you can then stop or keep continuing.

The more iterations, the more detailed the Julia set will look when zooming
in deeply, but the more calculations are needed. The higher the precision of the
numbers, the longer you can zoom in without encountering blocky pixels.

Here's the source code of a program that'll draw a Julia Set. Put this in the
main.cpp file, inside the "int main(int
argc, char *argv[])" function. The comments in it are made blue and
explain most of the code. The code doesn't use complex numbers but normal
floating point numbers, the real and imaginary part are simply calculated
separately, like you'd do by hand.

int main(int argc, char *argv[])
{
screen(400, 300, 0, "Julia Set"); //make larger to see more detail!
//each iteration, it calculates: new = old*old + c, where c is a constant and old starts at current pixel
double cRe, cIm; //real and imaginary part of the constant c, determinate shape of the Julia Set
double newRe, newIm, oldRe, oldIm; //real and imaginary parts of new and old
double zoom = 1, moveX = 0, moveY = 0; //you can change these to zoom and change position
ColorRGB color; //the RGB color value for the pixel
int maxIterations = 300; //after how much iterations the function should stop
//pick some values for the constant c, this determines the shape of the Julia Set
cRe = -0.7;
cIm = 0.27015;
//loop through every pixel
for(int x = 0; x < w; x++)
for(int y = 0; y < h; y++)
{
//calculate the initial real and imaginary part of z, based on the pixel location and zoom
and position values
newRe = 1.5 * (x - w / 2) / (0.5 * zoom * w) + moveX;
newIm = (y - h / 2) / (0.5 * zoom * h) + moveY;
//i will represent the number of iterations
int i;
//start the iteration process
for(i = 0; i < maxIterations; i++)
{
//remember value of previous iteration
oldRe = newRe;
oldIm = newIm;
//the actual iteration, the real and imaginary part are calculated
newRe = oldRe * oldRe - oldIm * oldIm + cRe;
newIm = 2 * oldRe * oldIm + cIm;
//if the point is outside the circle with radius 2: stop
if((newRe * newRe + newIm * newIm) > 4) break;
}
//use color model conversion to get rainbow palette, make brightness black if maxIterations reached
color = HSVtoRGB(ColorHSV(i % 256, 255, 255 * (i < maxIterations)));
//draw the pixel
pset(x, y, color);
}
//make the Julia Set visible and wait to exit
redraw();
sleep();
return 0;
}

The parameter of the number of iterations is used for the "Hue" of the HSV color
model. The advantage of Hue is that it's circular, so no matter how many maximum
iterations there are, the Hue based color palette will always keep generating
nice continuous values.

The result looks like this:

Julia Explorer

You can change the values "zoom", "moveX" and "moveY" in the above code to zoom
in at certain positions, but much better would be if you could do this in
realtime while the program is running, for example by pressing the arrow keys to
move, and the keypad + and - keys to zoom in/out. Even better would be if you
could use the keypad arrow keys to change the values of cRe and cIm to change
the shape of the Julia Set in realtime.

Programming something that can do that is pretty simple, just use the SDL keys
to change the values of those variables and draw the Julia Set every time again
in a loop! The following code does all that and also shows the values of all
those variables on screen, so that you know exactly at what coordinates of the
Julia Set the nice things are. It'll also let you change the value
"maxIterations" etc... The comments in the code should explain it all again.
There isn't any new computer graphics code, just input keys that change
parameters:

int main(int argc, char *argv[])
{
screen(320, 240, 0, "Julia Explorer");
//each iteration, it calculates: new = old*old + c, where c is a constant and old starts at current pixel
double cRe, cIm; //real and imaginary part of the constant c, determines shape of the Julia Set
double newRe, newIm, oldRe, oldIm; //real and imaginary parts of new and old
double zoom=1, moveX=0, moveY=0; //you can change these to zoom and change position
ColorRGB color; //the RGB color value for the pixel
int maxIterations=128; //after how much iterations the function should stop
double time, oldTime, frameTime; //current and old time, and their difference (for input)
bool keyReleased, keyPressed; //two bools for keys
int showText=0;
//pick some values for the constant c, this determines the shape of the Julia Set
cRe = -0.7;
cIm = 0.27015;
//begin the program loop
while(!done())
{
//draw the fractal
for(int x = 0; x < w; x++)
for(int y = 0; y < h; y++)
{
//calculate the initial real and imaginary part of z, based on the pixel location and zoom and
position values
newRe = 1.5 * (x - w / 2) / (0.5 * zoom * w) + moveX;
newIm = (y - h / 2) / (0.5 * zoom * h) + moveY;
//i will represent the number of iterations
int i;
//start the iteration process
for(i = 0; i < maxIterations; i++)
{
//remember value of previous iteration
oldRe = newRe;
oldIm = newIm;
//the actual iteration, the real and imaginary part are calculated
newRe = oldRe * oldRe - oldIm * oldIm + cRe;
newIm = 2 * oldRe * oldIm + cIm;
//if the point is outside the circle with radius 2: stop
if((newRe * newRe + newIm * newIm) > 4) break;
}
//use color model conversion to get rainbow palette, make brightness black if maxIterations
reached
color = HSVtoRGB(ColorHSV(i % 256, 255, 255 * (i < maxIterations)));
//draw the pixel
pset(x, y, color);
}
//print the values of all variables on screen if that option is enabled
if(showText <= 1)
{
print("X:", 1, 1, RGB_White, 1); print(moveX, 0, 17, 1, RGB_White, 1);
print("Y:", 1, 9, RGB_White, 1); print(moveY, 0, 17, 9, RGB_White, 1);
print("Z:", 1, 17, RGB_White, 1); print(zoom, 0, 17, 17, RGB_White, 1);
print("R:", 1, 25, RGB_White, 1); print(cRe, 0, 17, 25, RGB_White, 1);
print("I:", 1, 33, RGB_White, 1); print(cIm, 0, 17, 33, RGB_White, 1);
print("N:", 1, 41, RGB_White, 1); print(maxIterations, 0, 17, 41, RGB_White, 1);
}
//print the help text on screen if that option is enabled
if(showText == 0)
{
print("Arrows move (X,Y), Keypad +,- zooms (Z)", 1, h - 33, RGB_White, 1);
print("Keypad arrows change shape (R,I) ", 1, h - 25, RGB_White, 1);
print("Keypad *,/ changes iterations (N) ", 1, h - 17, RGB_White, 1);
print("a to z=presets (qwerty), F1=cycle texts", 1, h - 9, RGB_White, 1);
}
redraw();
//get the time and old time for time dependent input
oldTime = time;
time = getTicks();
frameTime = time - oldTime;
//no key was pressed yet
keyPressed = 0;
readKeys();
//ZOOM keys
if(inkeys[SDLK_KP_PLUS]) {zoom *= pow(1.001, frameTime); keyPressed = 1;}
if(inkeys[SDLK_KP_MINUS]) {zoom /= pow(1.001, frameTime); keyPressed = 1;}
//MOVE keys
if(inkeys[SDLK_DOWN]) {moveY += 0.0003 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_UP]) {moveY -= 0.0003 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_RIGHT]) {moveX += 0.0003 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_LEFT]) {moveX -= 0.0003 * frameTime / zoom; keyPressed = 1;}
//CHANGE SHAPE keys
if(inkeys[SDLK_KP2]) {cIm += 0.0002 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_KP8]) {cIm -= 0.0002 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_KP6]) {cRe += 0.0002 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_KP4]) {cRe -= 0.0002 * frameTime / zoom; keyPressed = 1;}
//keys to change number of iterations
if(inkeys[SDLK_KP_MULTIPLY] && keyReleased) {maxIterations *= 2; keyReleased = 0;}
if(inkeys[SDLK_KP_DIVIDE] && keyReleased) {if(maxIterations > 2) maxIterations /= 2;
keyReleased = 0;}
//key to change the text options
if(inkeys[SDLK_F1] && keyReleased) {showText++; showText %= 3; keyReleased = 0;}
//enable keyReleased again if the keys above are released
if(!inkeys[SDLK_KP_MULTIPLY] && !inkeys[SDLK_KP_DIVIDE] && !inkeys[SDLK_F1])
keyReleased = 1;
}
}

Now you can explore all the details of every possible Julia Set! Find a nice
shape with the keypad numbers, then move with the arrow keys to a border or
interesting spot of the julia set, and start zooming in to see more detail. If
you've zoomed in, you may want to press the numpad "*" button to see more
detail.

Note: in the current version of gcc, the pow function doesn't work if you mix
floats and doubles, so make sure either everything is double, or everything is
float.

Here are some screenshots:

A few different shapes:

Zooming in:

Increasing the number of iterations:

Mandelbrot Set

When zooming in, you'll keep seeing the same details over and over in a Julia
Set - it's a fractal after all. The Mandelbrot set isn't completely self
similar, it's only semi self similar, so in a Mandelbrot set much more surprises
can turn up when zooming in.

The mandelbrot set represents every complex point c for which the Julia Set will
be connected, or every Julia Set that contains the origin. To generate a
mandelbrot set you use the same iterative function as the Julia Set, only this
time c will represent the position of the pixel, and z will start at (0,0)

The following code is very similar to the Julia Set drawer, only it'll output a
Mandelbrot Set with rainbow palette instead. The changed parts are indicated in
bold.

int main(int argc, char *argv[])
{
screen(400, 300, 0, "Mandelbrot Set"); //make larger to see more detail!
//each iteration, it calculates: newz = oldz*oldz + p, where p is the current pixel, and oldz
stars at the origin
double pr, pi; //real and imaginary part of the pixel p
double newRe, newIm, oldRe, oldIm; //real and imaginary parts of new and old z
double zoom = 1, moveX = -0.5, moveY = 0; //you can change these to zoom and change position
ColorRGB color; //the RGB color value for the pixel
int maxIterations = 300;//after how much iterations the function should stop
//loop through every pixel
for(int x = 0; x < w; x++)
for(int y = 0; y < h; y++)
{
//calculate the initial real and imaginary part of z, based on the pixel location and zoom
and position values
pr = 1.5 * (x - w / 2) / (0.5 * zoom * w) + moveX;
pi = (y - h / 2) / (0.5 * zoom * h) + moveY;
newRe = newIm = oldRe = oldIm = 0; //these should start at 0,0
//"i" will represent the number of iterations
int i;
//start the iteration process
for(i = 0; i < maxIterations; i++)
{
//remember value of previous iteration
oldRe = newRe;
oldIm = newIm;
//the actual iteration, the real and imaginary part are calculated
newRe = oldRe * oldRe - oldIm * oldIm + pr;
newIm = 2 * oldRe * oldIm + pi;
//if the point is outside the circle with radius 2: stop
if((newRe * newRe + newIm * newIm) > 4) break;
}
//use color model conversion to get rainbow palette, make brightness black if maxIterations
reached
color = HSVtoRGB(ColorHSV(i % 256, 255, 255 * (i < maxIterations)));
//draw the pixel
pset(x, y, color);
}
//make the Mandelbrot Set visible and wait to exit
redraw();
sleep();
return 0;
}

Isn't that pretty:

Mandelbrot
Explorer

Here's again the full code of a program that allows you to move around and zoom
in the Mandelbrot Set.

int main(int argc, char *argv[])
{
screen(320, 240, 0, "Mandelbrot Explorer");
//each iteration, it calculates: new = old*old + c, where c is a constant and old starts at current pixel
double pr, pi; //real and imaginary part of the pixel p
double newRe, newIm, oldRe, oldIm; //real and imaginary parts of new and old
double zoom = 1, moveX = -0.5, moveY = 0; //you can change these to zoom and change position
ColorRGB color; //the RGB color value for the pixel
int maxIterations = 128; //after how much iterations the function should stop
double time, oldTime, frameTime; //current and old time, and their difference (for input)
bool keyReleased, keyPressed; //two bools for keys
int showText = 0;
//begin main program loop
while(!done())
{
//draw the fractal
for(int x = 0; x < w; x++)
for(int y = 0; y < h; y++)
{
//calculate the initial real and imaginary part of z, based on the pixel location and zoom
and position values
pr = 1.5 * (x - w / 2) / (0.5 * zoom * w) + moveX;
pi = (y - h / 2) / (0.5 * zoom * h) + moveY;
newRe = newIm = oldRe = oldIm = 0; //these should start at 0,0
//i will represent the number of iterations
int i;
//start the iteration process
for(i = 0; i < maxIterations; i++)
{
//remember value of previous iteration
oldRe = newRe;
oldIm = newIm;
//the actual iteration, the real and imaginary part are calculated
newRe = oldRe * oldRe - oldIm * oldIm + pr;
newIm = 2 * oldRe * oldIm + pi;
//if the point is outside the circle with radius 2: stop
if((newRe * newRe + newIm * newIm) > 4) break;
}
//use color model conversion to get rainbow palette, make brightness black if maxIterations reached
color = HSVtoRGB(ColorHSV(i % 256, 255, 255 * (i < maxIterations)));
//draw the pixel
pset(x, y, color);
}
//print the values of all variables on screen if that option is enabled
if(showText <= 1)
{
print("X:", 1, 1, RGB_White, 1); print(moveX, 0, 17, 1, RGB_White, 1);
print("Y:", 1, 9, RGB_White, 1); print(moveY, 0, 17, 9, RGB_White, 1);
print("Z:", 1, 17, RGB_White, 1); print(zoom, 0, 17, 17, RGB_White, 1);
print("N:", 1, 25, RGB_White, 1); print(maxIterations, 0, 17, 25, RGB_White, 1);
}
//print the help text on screen if that option is enabled
if(showText == 0)
{
print("Arrows move (X,Y), Keypad +,- zooms (Z)", 1, h - 25, RGB_White, 1);
print("Keypad *,/ changes iterations (N) ", 1, h - 17, RGB_White, 1);
print("a to z=presets (qwerty), F1=cycle texts", 1, h - 9, RGB_White, 1);
}
redraw();
//get the time and old time for time dependent input
oldTime = time;
time = getTicks();
frameTime = time - oldTime;
//no key was pressed yet
keyPressed = 0;
readKeys();
//ZOOM keys
if(inkeys[SDLK_KP_PLUS]) {zoom *= pow(1.001, frameTime); keyPressed = 1;}
if(inkeys[SDLK_KP_MINUS]) {zoom /= pow(1.001, frameTime); keyPressed = 1;}
//MOVE keys
if(inkeys[SDLK_DOWN]) {moveY += 0.0003 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_UP]) {moveY -= 0.0003 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_RIGHT]) {moveX += 0.0003 * frameTime / zoom; keyPressed = 1;}
if(inkeys[SDLK_LEFT]) {moveX -= 0.0003 * frameTime / zoom; keyPressed = 1;}
//keys to change number of iterations
if(inkeys[SDLK_KP_MULTIPLY] && keyReleased) {maxIterations *= 2; keyReleased = 0;}
if(inkeys[SDLK_KP_DIVIDE] && keyReleased) {if(maxIterations > 2)
maxIterations /= 2; keyReleased = 0;}
//key to change the text options
if(inkeys[SDLK_F1] && keyReleased) {showText++; showText %= 3; keyReleased = 0;}
//enable keyReleased again if the keys above are released
if(!inkeys[SDLK_KP_MULTIPLY] && !inkeys[SDLK_KP_DIVIDE] && !inkeys[SDLK_F1])
keyReleased = 1;
}
return 0;
}

By moving around and zooming you can get very pretty pictures here. The
parameter values were left on the pictures so you can see the coordinates, the
zoom level and number of maximum iterations to generate those pictures:

This part of the Mandelbrot Set is called the Seahorse Valley:

A deeper zoom to the size of the Seahorse Valley (zoomed in 1779 times):

A detail on the other side of the set:

A few "minibrots", the first zoomed in only 8463 times, the second zoomed in
419622325484 times! Such minibrots only become visible if you allow enough
iterations.

This is the effect of using more iterations: on the right is the same picture on
the left, but with twice as much maximum iterations:

This is a deep zoom with a very high number of iterations in the side of the
elephant valley. The elephant valley is the pointy shape on the X-axis, on right
side of the mandelbrot.

This picture is added more recently, because it's pretty pretty:

And here's a picture that's zoomed in so much, that the numerical limit of
64-bit floating point numbers was reached. It starts looking blocky. It's zoomed
in more than 10^18 times though, for deeper detail, processors with higher
precision, or emulation of infinite precision numbers is required: