I didn't why they got hung up, but I'm not a mathematician.
Given the information they have, it seems they can, starting with the 2D Julia set, use an evolutionary procedure to fold the object into a 3D shape (potentially not unique) such that each point on the Julia set has the correct MME. i.e., you try a random fold and have a particle follow many random walks, and measure the MME at each point. If the MMEs are closer to their true values, you keep the fold. After this is all done, apply a smoothing algorithm that minimizes the number of folds.
I think that's a neat idea. More specifically, I think they could have used it to replace this step:
> Today, the best strategy is often to make a best guess about where to fold the polygon — and then to get out scissors and tape to see if the estimate is right.
“Kathryn and I spent hours cutting out examples and gluing them ourselves,” DeMarco said.
My understanding is that that was something they did to get some intuition on where these folds occur, which would hopefully aid in spotting a pattern of some sort.
It seems like the critical thing is that there is some underlying pattern to where the folding lines occur, though—otherwise it's not really of theoretical interest. Something like this seems like their end goal:
> “Certain polynomials might have similar bending laminations, and that would tell us all these polynomials have something in common, even if on the surface they don’t look like they have anything in common,” Lindsey said.
And it seems like they have made some headway on understanding the underlying 'pattern':
> “Our working conjecture is that the folding lines, the bending laminations, can be completely described in terms of certain dynamical properties,”
But I bet if they'd taken your approach rather than manually cutting things out of paper, they'd have much better data for thinking about this.
(Although, looking closer at your proposed algorithm, I'm guessing it would need some modification: the folding shouldn't affect the MMEs—they have a curvature distribution derived from the MME already, but:
> If given a two-dimensional polygon, and told exactly how its curvature should be distributed, there’s still no mathematical way to identify exactly where you need to fold the polygon to end up with the right 3-D shape.
So I think the different folding schemes are independent of MME/curvature distribution.
Still, something along the lines of what you described might work...)
Edit: I should point out that I'm also not a mathematician :D
Given the information they have, it seems they can, starting with the 2D Julia set, use an evolutionary procedure to fold the object into a 3D shape (potentially not unique) such that each point on the Julia set has the correct MME. i.e., you try a random fold and have a particle follow many random walks, and measure the MME at each point. If the MMEs are closer to their true values, you keep the fold. After this is all done, apply a smoothing algorithm that minimizes the number of folds.