Is The *Mona Lisa* in the complement of the Mandelbrot set. 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 $2$.

I think yes
Consider the sequence of "westernmost" islands increasing in period:

Here are some examples, you can see them increasing in hairyness / spinyness.
Period 20:

Period 30:

Period 40:

Period 50:

Zooming in near to a sufficiently hairy high-period island, you can get very nearly parallel spines:

There is a natural grid with a 4:1 aspect ratio between successive escape time level sets in neighbouring spine gaps.  (Determined by measuring images: I have no proof of this seeming fact.  But it's not a strictly necessary detail.)
If you choose the angle of the spines carefully, you can create an NxN grid with square cells that satisfies "distinct level set at each pixel sampling point in an (m,n) image array".

There is a slight fuzziness due to the imperfect parallelism and the quantized angles available for any given period island, but increasing the period far enough and that won't matter - eventually it gets good enough for the finite width of the level sets to take care of it.
