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The equations/algorithms of that define attractors such as these:

With given parameters and number of iterations output a set of positions, the size of the set is the same as the number of iterations.

Can attractors like these, or their $2D$ versions be rearranged to simulate an infinite number of iterations?

The new equation/algorithm would take the parameters and additionally $X, Y$ and, if applicable, $Z$ coordinates, but no number of iterations.

Returning a non-negative real number instead of a set of positions, this number would represent the distance between the specified position and the nearest point in the attractor.

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"but no number of iterations." - why? All you can really do is a finite number of iterations and thus only an approximation of the true attractor. – J. M. May 16 '11 at 12:27
As covered before I think it should be possible to rearange this problem so that you find the proximity to the nearest part of the infinite set (obviously not at any point involving returning the infinite set) instead of finding a finite sized chunk of the start of the set. If you think this is not possible please answer as such. After some time I can then except your answer untill someone solves it, or dosn't if isn't possible. Also thanks for the tag. – alan2here May 16 '11 at 12:44
I don't have time to elaborate (and thus am just leaving my thoughts as comments), but due to finite precision among other things, we don't know what the true "infinite iterations" attractors actually look like, and thus those graphics are mere approximations of those attractors (and we're hoping that they're a good approximation). It's a bit hard to calculate "proximity" of something approximate if you don't know what the exact thing is like... – J. M. May 16 '11 at 12:56

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