Short Explanation
The Mandelbrot Set is the result of some computation performed on every point in a plane. For each point, the result is a number, which only depends on the coordinates of that point. This number is transformed into a color, tyically by using a list of predefined colors (the colormap).
Example coordinates Coordinates


Every point in the picture can be identified using two numbers, the distance from the origin to the right and the distance downwards, or using negative values left- and upwards. The origin is (0,0) and the other blue point in the picture is (-1, -0.5). To describe the area the picture contains, you just give the coordinates for the upper left corner [here: (-2.5, -1.5)] and the lower right corner [here: (1.5, 1.5)].
Start

So lets start with an image that has in its upper left corner the coordinates (-2.5, -1.5) and in its lower right corner (1.5,1.5). These are the values used for the image above.
Now we have for every point the coordinates (x, y), where in this example x ranges from -2.5 to 1.5 and y ranges from -1.5 to 1.5, we can compute a value that will define the color, using the following algorithm.
Algorithm

Complex Numbers
If you know about complex numbers, the computation looks pretty simple.
Each point in the final image defines a complex number, i.e. for the coordinates (x, y) you get the number a0=x+i*y. Then do the following:
  1. Having an (starting with n=0), you can compute an+1=an2+a0
  2. If |an| ≥ 4, look up the n-th color in your colortable and draw this point with this color.
  3. If |an| stays below 4 forever, color the point black.
    How do we know that it will never grow above 4? We don't, so we have to guess: Whenever it didn't get larger than 4 until n reaches 100 (or 1000, 5000, ...), we assume it never will.Choosing this maxiter-value can be difficult, therefore the MandelApplet tries to do it automatically.
  4. If n hasn't yet reached the maxiter-value and |an| < 4, go back to step 1.
Without complex numbers
Without using complex numbers, the computation looks slightly more complicated.
If you want to know the color of a point having coordinates (x,y), you must do the following:Start the computation with a=x, and b=y. Now:
  1. Compute a'=a2-b2+x, and b'=2*a*b+y
  2. Replace a with a', and b with b'
  3. If sqrt(a2+b2) < 4 (or a2+b2 < 16), go back to step 1.
  4. Otherwise paint the point with a color corresponding to the number of times you had to repeat steps 1-3.
  5. For some points, you would have to repeat these steps infinitely, so you need a maxiter-value that limits the number of repetitions. If you reach maxiter repeatitions, color the point black.
Remarks
Of course you can choose a different value than 4 for the comparison, or use some other color than black for those points where the computation never grows above 4, but these are the typically used values.
Tips and Features
Fractint.org Start with the autopilot. The Applet has an autopilot that will take you deeper and deeper into the Mandelbrot set, as far as the normal arithmetic of your PC goes. The autopilot examines the computed image and zooms into a region where it finds high contrasts.
MaxIter Icon Automatic MaxIter selection. To compute a mandelbrot set one has to choose an apropriate value for maximum iterations. Higher values will take longer to compute, and lower values may result in incorrect images. The Applet tries to do this automatically based on the last computed image. Sometimes this prediction will be too low and then you will see an image with "missing" interiors. Just go on and let the automatic catch up.
mandelbrot icon Déjà vu. Mandelbrot sets have a property called self-similarity. In other words, you will find similar structures again and again while zooming in. Look out for occurences of the starting image.