[messe]

> These kind of fractal have actually 4 dimentional structure since c_x and c_y can also be parameters

They're not parameters in that sense.

The fractal is computed by taking each point on the plane as coordinates (c_x, c_y), and then iteratively applying the recursion relation. Then, with luminosity depending on how quickly that sequence escapes to infinity, we color in that point (c_x, c_y) in our image.

[ithinkso]

That's not what kenshoen meant. For example for the Mandelbrot set we have a function f(z) = z^2 + c for complex z,c and each pixel in the image represents c_x,c_y in c = c_x + i*c_y and then you iterate f(0), f(f(0))... On the other hand, if you have c constant and each pixel represents z_x, z_y in z=z_x + i*z_y instead, then iterating f(z),f(f(z))... gives you a Julia set.

But you can think of f as a function of two complex arguments f(z,c)=z^2 + c and iterate it on the whole domain (two complex = four real dimensions) and then have a picture being a slice through any 2D or (even 3D, which is what parent is talking about) plane you like. In other words, the famous Mandelbrot fractal picture is a slice of f(z,c) through a plane z=0, and Julia set pictures are slices through planes c=constant but there is no reason one cannot make other pictures of f(z,c) (just be careful what you meant by iterating a function f: C^2 -> C).

The burning ship fractal in the article is the same but the function f(z,c) is a bit weirder