What determines alpha on those images? On any attractors really? You end up with an infinitely large set of points between say -1.5 and 2.0. How does that end up being graphed. Perhaps that is magic/beauty of it.
1) Choose a target image resolution, and set up a square array of that size, initialized to zeroes.
2) Decide on the bounds you want to use, and linearly map coordinates into the target image. Say you want to graph x in the range [Xmin, Xmax) and your resolution is Xres, then pixel_x = (int)((x - Xmin) * Xres / (Xmax - Xmin)) (Note that if you have a higher-dimensional attractor you'll need some form of projection down to two dimensions.)
3) Iterate your attractor equation (starting point doesn't matter because it's an attractor). Each iteration, increment the pixel that matches your current coordinates. Stop iterating when you land on a pixel with a value above some preset.
4) Apply a palette to your pixel values to turn it into a nice picture like the ones linked here.
I wrote a quick JS vesrion using the provided formulas: http://bl.ocks.org/gunn/87401cb547cc2ee5b5ec
I know alpha refers to many things but one of those is the fine-structure constant. There's some remarkable beauty in it because it unites electromagnetism, relativity and quantum mechanics. And this is why many thought the reciprocal of alpha (137) would become as important as 0 and Pi.
I am assuming though none of that relates to fractals and that alpha in this context refers to the angles?
I quickly wrote something that would output the numbers and mapped those to points on the screen. But it just ended up looking like that. Points on a screen.
The images on the site looked like they had alpha... or used a much better idea than mine of just using points on a plane.