I was looking at the Menger Sponge earlier, and I realized it has a neat property:
Let x, y, and z be spatial dimensions, each between 0 and 1 (inclusive.) Express them as ternary floating point numbers. (x,y,z) belongs to the interior if and only if none of them contain the digit 1.
If you operate in ternary from the outset, you can 'immediately' tell if (x,y,z) is included in the interior - but in general you still have to examine n digits to gain just as much precision. If all three variables are rational, you have only a finite number of digits to examine - maybe it's possible to take advantage of that?
Anyway, I'm almost certain that the Mandelbrot set can't be pinned down with a closed form, or anything that can be described as 'chaotic'. Are there any that can? If so, what distinguishes them from those that can't?
I get the feeling I'm on the verge of running into a whole field of mathematics - maybe real/complex analysis? I just don't have the experience to recognize it when I see it.
Edit - maybe with the Menger Sponge, the remaining 2 ternary digits (0 and 2) can still occur without repeating, which makes them irrational, so a closed form would have to take infinitely many digits into account. Am I looking in the right direction?
Your observation is incorrect. The set of points $(x,y,z)$ where none of them have $1$ in ternary expansion is the product of three Cantor sets. This is a rather small subset of the Menger sponge: it has smaller dimension, and is totally disconnected.
You can still describe the Menger sponge in terms of ternary expansions, but they have to be considered together. To see why, begin with the simpler case of the Sierpinski carpet.
Yes, some fractals have neat expressions in coordinates than others, while others, like Mandelbroit set and Julia sets, do not. I'm not sure this means one is somehow more complex than the other: it's just that certain tools are better suited for certain objects and not for others. For example, it's easy to find the logarithmic capacity of the Julia set of a polynomial, while for the standard Cantor set this is quite a challenge.