What's the points?

Back in January 2007, I was browsing Paul Bourke's web site, looking at amazing images generated from fractals. Like most people I find that some of them have an artsy side and would fit perfectly in a modern art exhibit.

Then I came upon this one:

$ x_{n+1}=\sin(ay_n)-\cos(bx_n) $
$ y_{n+1}=\sin(cx_n)-\cos(dy_n) $

It was generated with a mathematical attractor named 'Peter de Jong'. It was simple, no colors and the rendering wasn't that great. Even though all the points are in a single 2D flat plane, I couldn't help seeing depth in this image. To me, there's a bunch of translucent surfaces in a 3D space.
Of course, at that point my brain started working on overdrive wondering what it would take to separate the points for real and see the shape expanding into a 3rd dimension. So I've set some basic rules for this experiment:

1) Only use the existing X and Y data to generate the new Z data.
2) Never modify the existing X and Y data.

I called this process delamination, because it's like trying to separate different layers glued on top of each other. So far I've achieve 80% to 90% delamination. I only understand 20% of the process and I'm analyzing my results to figure it out.

Also. If you've worked on something like this, I would really like to talk with you.