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The equations/algorithms of these attractors (the link below) for given parameters and a number of iterations output a set of positions, the size of the set is the same as the number of iterations:


Take, for example, pickover some way down the page. $X,Y,Z$ start at some values. The $X, Y,Z$ with the dots over them mean $X, Y, Z$ in the next iteration, on the left is a picture of what this looks like after many iterations.

Can they 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. It would return a single number instead of a set of positions, this number would represent the distance between the specified position and the nearest point in the attractor.

The number returned should never be negative but sometimes it may be exactly 0 where either a point has been chosen that is exactly on top of an existing point or the point in the attractor jumps chaotically, or in the case of some fractal attractors systematically possibly covering all the space in some part of 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
My question is "how close does the attractor ever come to this position" instead of "where are the first few position that the atractor reaches" and is thus a specific case of a more general question I asked before where it was concluded that no generic solution existed to that problem, for the same reason I suposed that no generic solution existed amoung all attractors but that individual ones could be found for individual attractors. –  alan2here May 16 '11 at 13:25

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