3-D Fractals Offer Clues to Complex Systems
quantamagazine.org
quantamagazine.org
> “The Julia set is the base, like the southern hemisphere, and the cap is like the top half,” DeMarco said. “If you glue them together you get a shape that’s polyhedral.”
And the cap comes from:
> To get the second, DeMarco and Lindsey wrote an algorithm. That algorithm analyzes features of the original polynomial, like its degree (the highest number that appears as an exponent) and its coefficients, and outputs another fractal shape that DeMarco and Lindsey call the “planar cap.”
It may just be that the article doesn't present this aspect clearly enough—but from the description given, it just makes me wonder: why would it be interesting to glue these two pieces together?
I suppose if their geometry is perfectly complementary so that all the complex fractal whorls fit together like puzzle pieces, then that's pretty interesting and surprising, and a key point—but the article doesn't touch on it!
Also, (again, from the description given by the article), these '3-D fractals' arrived at by folding and gluing together planar fractals, seem fundamentally less interesting than the other 3D fractals which are more deserving of the name, IMO (e.g. the Mandelbulb: https://en.wikipedia.org/wiki/Mandelbulb).
Then again, they're probably more interested in discovering deep properties about polynomials than making pretty things, unlike myself, so... go them.
Also don't forget that the researchers themselves originally didn't understand its significance when Thurston suggested the idea.
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.
> 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
however one problem is always requires computing all the backwards iterates of the points. And there's never getting around that.
these two ladies offer a means of approximating the Julia set using a polygon. I have no heard of this.
there is a nice result by William Thurston they are using "Shapes of polyhedra and triangulations of the sphere" https://arxiv.org/abs/math/9801088
The theorem is exactly what the article says. They cut out this shape and fold it along the correct lines and glue along the boundary, one obtains the boundary of a convex shape.