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"what they’ve done is set up three screens, in a triangular pattern"

Triangular pattern doesn't say very much. Is this supposed to mean that the trick will work iff the screens are not collinear?

You can do something similar, although not quite as intricate, by simply pointing your webcam at a computer screen that is displaying the webcam-feed. You get infinitely repeating patterns that respond in beautiful ways to subtle tilts and nudges of the webcam. It's really pretty impressive and worth 5 minutes of your time if you have an external webcam around.
Now all you need is an app that shows the video feed in multiple windows you can drag around and you can get the exact same effect.
Or with Mac OS X screen sharing, VNC, or similar.
Personally, i think this speaks to the way time works - but i've done A LOT of mushrooms....
Technically most fractals occur without a computer.
Like when they were discovered!
Doesn't it depend on your precise definition of Fractal. In nature they're not self-similar repetitions at altered scale but approximate the same¹. There is always a limitation in nature of resolution (lumpiness of atoms) just as no true circle can exist in nature (at least in our realm Plato², please correct me if I'm wrong).

So technically most (all I think) fractals do not occur extent in reality but only the mathematical description of them occurs.

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¹ note the precise definition of "similar" in geometry, http://en.wikipedia.org/wiki/Similarity_%28geometry%29. ² I can't recall Plato's take on [perfect] forms of biological "shapes" like trees and such.

_Godel, Escher, Bach_ describes similar phenomena with camera feedback loops. When I read GEB, I wondered if it would be possible to create a Sierpinski triangle using four specially shaped/curved mirrors. I briefly played with povray, but didn't figure it out.
Does real-time video not at the very least involve a micro-controller? So arguably this is one way to make fractals WITH a computer. It looks cool and video feedback does make nice effects.