Mandelbrot graphs
The Mandelbrot set, first visualized in the early 1980s by Benoit B. Mandelbrot is simply the set of complex numbers c for which an iterative evaluation of
z_{n+1} = z_{n}^{2} + c
where z_{0} = c
will never approach infinity for any n, that is, will never leave a circle around origo with a radius of 2.
Now, if you are bothered by complex numbers being nonintuitive, welcome to the club. Just view them as coordinates (x, y) in a coordinate system.
To perform this calculation, one needs to choose a "bailout value"  a max value of n in the formula, to be able to terminate the calculation for those points that are inside the set  otherwise there would be no way to stop calculating. This is computationally demanding stuff, and one will have to choose this bailout value carefully.
Or had to, at any rate. On today's computers you can probably set it to whatever you feel like.
This is an "in or not in" situation where the basic mathematical presentation is in the form of a b&w graph with the Mandelbrot set coloured black.
Rather boring stuff, in other words  unless you are sensible enough to be fascinated by the fact that this set will present infinite detail along the edges no matter how far you zoom into it, or by the fact that the edge of the set is single and unbroken. This edge, by the way, is of infinite length, even though the area of the mandelbrot set is finite
Now, the Mandelbrot graph can easily be turned into a colourful, psychedelic picture by simply assigning a colour to each point outside the Mandelbrot set proper, the colour chosen depending on how many iterations of the above formula it takes before z has left the r=2 circle. This is the kind of "Mandelbrot graph" you will usually see.
Actually, this does present some problems since you will have to define a colour table of a suitable size to match the actual set of n values in the actual subset of the graph that you are currently visualizing, and also depending on your current bailout value. Each colouring model will present a subset in a different light.
Note: I've never tried using the Normalized Iteration Count Algorithm or anything like that to smooth out the colouring, since I consider it almost cheating. I've concentrated on improving the colour schemes without deviating from the normal, simple Escape Time algorithm.
In the menu you can find several applets to try out.
Change in formula
In the past, I've always used the formula that Benoît Mandelbrot himself published, "z > z^{2}  c". However, the popular version of the formula nowadays is "z > z^{2} + c", the only practical difference being the orientation across the y axis. Some of my images may still be made with the former formula. I'm in the process of tidying up my collection and converting old dumps but it's not ready yet.
I posted the following question on the Talk:Mandelbrot set page on Wikipedia in 2006, and received a good reply:
The formula as originally presented by BBM was z > z^{2}  c but almost every single current reference uses z > z^{2} + c. Anybody know when and why this was changed? Khim1 14:23, 17 January 2006 (UTC)
 Mandelbrot's contribution to The Beauty of Fractals (Peitgen and Richter; 1986) refers to "the quadratic map z > z^{2}  c". In 1988 Michael Barnsley uses z > z^{2}  λ in Fractals Everywhere. OTOH Peitgen and Richter's 1986 survey article Frontiers of Chaos, also published in The Beauty of Fractals, uses the map x > x^{2} + c and includes illustrations of the Mandelbrot set in its "modern" orientation. I don't know why the second form became more popular. Gandalf61 11:55, 23 January 2006 (UTC)

 I am not sure about the historic development, but certainly Douady and Hubbard, in the early 80's, defined the Mandelbrot set in its current form. In this parametrization, the parameter c agrees with the singular value of the map , which is useful from a conceptual point of view. LR 22:56, 17 March 2006 (UTC)
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