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I've recently had nice results with concentric circles folded a square pattern, to emulate a drop falling in a square pond, except i made the drops bounce back to the centre from a square pond with variable size. The patterns are surprising variable, and they make me think of the mandelbox because the patterns are similar.

Is there a 2d mandelbox formula with a similar dergree of variation as teh 3d one, that can be zoomed in like a mandelbrot set? that would be cool because the patterns generated by squares and circles are extremely diverse even in just one iteration. it seems a cool avenue for research, circles folded onto squares. I know there are already some versions, could provide some web pages for them? if circles are folded over a square many times from different distances, the patterns could be interesting.

here is the waves emulator: <a href="http://www.youtube.com/v/hP4EwAPNVw0&rel=1&fs=1&hd=1" target="_blank">http://www.youtube.com/v/hP4EwAPNVw0&rel=1&fs=1&hd=1</a>

There is definitely some sort of similarity between a 2d Mandelbox and concentric rings bounced against a square box. This is because the box fold is a reflection around the box, which is exactly what a circular wave bouncing off a square box produces. The expansion of the wave is sort of represented by the scale factor in the Mandelbox equation.
However, it isn't exactly the same... a scale factor isn't the same as an enlargement of a ring by a fixed amount and the Mandelbox also includes a +C... but there is definitely similarity.

I think what your video is generating is standing waves... which is interesting in itself, of great interest to quantum theory for instance.

I agree with the general idea that invariant sets using the combination of linear and circular operations are interesting.

Thanks! very nice fractal and interesting, yes indeed, there are surely many possibilites with circles and cubes. i wrote a program for graphics shader to be able to see the standing wave emulator. here is the page, it needs a shader model 3 graphics card.
https://dl.dropboxusercontent.com/u/114667999/Public.html