Fractals are complicated images, whose parts are similar to the whole. In other words, if you zoom into a fractal, you can get a picture similar to the one you started with. Keep zooming in and again you’ll get a similar picture. And so on – to infinity!

This tutorial will let you quickly (only about 25 lines of code!) create the famous fractal called the Mandelbrot set.

Here is some basic math behind the concept. Feel free to skip the gray section if you’re more interested in the code than in the math.

A point in the complex plane belongs to the Mandelbrot set if the orbit of 0 under iteration of the quadratic map:

z_{n+1}=z_{n}^{2}+c

remains bounded (does not escape to infinity).

It can be plotted on a square image between -2-2i and 2+2i.

For practical reasons, we’ll only iterate 255 times for each point. We will only iterate a given point as long as the absolute value of z_{n} is lower than 4. If it’s 4 or greater, we know it will eventually escape to infinity, so we don’t have to continue the iterations. The number of iterations determines the point’s color.

It all boils down to the following:

0. Select the point in the bottom left corner of the region (-2,-2)

1. Start with zx=0 and zy=0

2. Calculate:

xt=zx*zy

zx=zx*zx-zy*zy+cx

zy=2*xt+cy

These calculations are really the formula z_{n}=z^{2}_{n+1}. The reason they look different is because i^{2}=-1, a property of complex numbers.

3. Repeat step 2 if:

a. you haven’t reached 255 iterations yet [i255]

AND

b. the absolute value of the result is lower than 4 [zx*zx+zy*zy)4]

4. change the color of the point you’re calculating (ie. cx,cy) to the color corresponding to the number of iterations (the number of times you repeated steps 2 and 3)

We set Red, Green and Blue values of the color of the square to the number of iterations, which generates a greyscale picture.

5. Select the next point (go by columns and rows)

6. Go to step 1, until you reach the bottom right corner

Here’s the actual code:

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<html> |

<body> |

<canvas id=”myCanvas” width=”800” height=”800“></canvas> |

<script> |

var canvas = document.getElementById(‘myCanvas‘); |

var context = canvas.getContext(‘2d’); |

for(var x=0;x<200;x++) |

{ |

for(var y=0;y<200;y++) |

{ |

var i=0; |

var cx=-2+x/50; |

var cy=-2+y/50; |

var zx=0; |

var zy=0; |

do |

{ |

var xt=zx*zy; |

zx=zx*zx–zy*zy+cx; |

zy=2*xt+cy; |

i++; |

} |

while(i<255&&(zx*zx+zy*zy)<4); |

var color=i.toString(16); |

context.beginPath(); |

context.rect(x*4, y*4, 4, 4); |

context.fillStyle =’#’+color+color+color; |

context.fill(); |

} |

} |

</script> |

</body> |

</html> |

__For beginners__: you can copy the code above, paste it into a text file (in Notepad), save it as ‘mandel.htm’ (HTML file) and open it in your browser. Your screen should look like this:

This HTML file is also available HERE. If you click on this link, a new window will open with the image of the set generated by the code.

It works and I think it looks really cool – a relatively simple math formula creates something infinitely complicated to the human eye. But it’s just the beginning! The real beauty of Mandelbrot only becomes apparent as you apply colors and zoom inside them to explore its endless lakes and caves.

I concentrated on keeping the code short, at the expense of the ‘cool factor’. Here’s where it gets more interesting:

If you don’t feel like reading another tutorial, you can just experiment with different values in these two lines:

cx=-2+x/50;

cy=-2+y/50;

For example these values:

cx=-.68+x/1200;

cy=-.74+y/1200;