The mandelbrot set is generated by repeatedly evaluating a function until it overflows (some defined limit), then checking how long it took you to overflow.
Pseudocode:
MAX_COUNT = 64 // if we haven't escaped to infinity after 64 iterations,
// then we're inside the mandelbrot set!!!
foreach (x-pixel)
foreach (y-pixel)
calculate x,y as mathematical coordinates from your pixel coordinates
value = (x, y)
count = 0
while value.absolutevalue < 1 billion and count < MAX_COUNT
value = value * value + (x, y)
count = count + 1
// the following should really be one statement, but I split it for clarity
if count == MAX_COUNT
pixel_at (x-pixel, y-pixel) = BLACK
else
pixel_at (x-pixel, y-pixel) = colors[count] // some color map.
Notes:
value is a complex number. a complex number (a+bi) is squared to give (aa-b*b+2*abi). You'll have to use a complex type, or include that calculation in your loop.