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Here is the dot

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And here is how I arrived at it »

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Pretty simple IFS using a 2 x 2 grid as the base for the iterations. Is there a way to describe this point as well as its siblings, all of which lie on the dotted red line(pic 2), using an equation?

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    The algorithm is pic 2. Here's the code » http://www.mediafire.com/?lazew4xvyc08wpp It uses 2, 1, 3 as the default input; enter your own sequence and see the results. More info in the file. There's no math involved as you can see from the file's html code. Tested in Chrome but should work in other browsers. The center of the grids that are affected by the sequence is the protagonist. The equation I'm looking for is its path as it moves with each iteration so I can retrieve, at a later date, it as well as its predecessors who were on the path, simply by plugging in simple integers into the eqn.2010-10-01

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It seems like you are applying $f_1,f_2,f_3$ where 1,2,3 maps the entire square to down right, up right, and down left respectively, and your red dot is the solution to $f_1(f_2(f_3(x))) = x.$ Now, you just have to find these functions, which is rather easy and left as an exercise. Now, the composition of these 3 functions (in the order above) is:

$f(x,y) = ((x+5)/8,(y-3)/8)$

Now, solve $(x+5)/8 = x$, giving $x=5/7.$ Now, solve $(y-3)/8 = x$, giving $x=-3/7.$

The red dot should be $(5/7,-3/7).$

I have assumed that the corners of the original square is $(\pm 1,\pm 1)$.

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It's not very clear what you're trying to ask, but what you're describing looks like a region quadtree.

Each step in your IFS corresponds to a choice of child node (or subquadrant), and the tree path of any given point is the same as the list of numeric input steps in your question.

It does not really matter whether you store the point as an arbitrary-precision coordinate pair or tree traversal: the binary digits required to store its coordinates correspond exactly to its quadtree traversal path.