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Here is a description of how to color pictures of the Mandelbrot set, more accurately the complement of the Mandelbrot set. Suppose we have a rectangular array of points. Say the array is $m$ by $n$. Suppose also we have a number of color names. Now suppose we assign the color name $j$ to a point in the array if the $j$-th iteration exceeds $2$. If the iterates do not exceed $2$ we color the point black. This process will yield a picture. By careful positioning the array of points can we get any picture we want? In particular can we get a digital representation of the Mona Lisa.

I do not know how to begin to prove or disprove this. My guess is that we can probably get any pictures.

Edit

A different way to color the array would be to use color $c$ if the first iterate to exceed $2$ is iterate $i_{1}$, $i_{2}$, $\cdots$, $i_{j_{c}}$. The iterates for different colors should be distinct. If someone wishes to use infinites lists for the number of iterates that are assigned to a color that would also be acceptable.

With this change the problem reduces to finding an $m$ by $n$ array where each point in the array has a different number of iterates before the value exceeds $2$.

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It appears you are envisioning a two-dimensional analog of the idea that eventually one can find any arbitrary sequence of digits in an infinite sequence with a uniform random distribution. The question, however, does not seem clearly formed. Most importantly, if the escape values in your array are unique, how can they generate the Mona Lisa whose array must contain identical values? If, as you see it, the Mona Lisa's array does not contain identical values, then this is akin to claiming that, for example, a 1000 x 1000 array of the numbers 1 to 10,000 is equivalent to the Mona Lisa. –  Harlan Aug 7 '13 at 6:25
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I assume that initially we have an $m$ by $n$ array of color names. This is how we choose to represent the Mona Lisa. Some points in this array will likely be assigned identical color names. The idea is to find an $m$ by $n$ array of points in the complement of the Mandelbrot set and a method of assigning color names to escape values(thanks for the terminology) so that the $m$ by $n$ array, when colored by this assignment, looks like the Mona Lisa. To be specific, points with the same escape value will have the same color. –  Jay Aug 7 '13 at 14:16

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