Trying to understand the DeMarco and Lindsey 3d folded fractals

Quanta magazine’s article on 3d folded fractals from last month has really captured my imagination:

3-D Fractals Offer Clues to Complex Systems

Since reading the article I’ve been trying to understand the paper by Laura DeMarco and Kathryn Lindsey that inspired the story:

Convex shapes and harmonic caps on arXiv.org

Although I’m making progress digesting the paper, that progress is slow – who knew that trying to understand current research in a field you know nothing about would be so hard . . . 🙂

One really nice thing in the paper that helped me get my bearings was figure 1.1:

screen-shot-2017-02-05-at-1-34-15-pm

This figure shows the curved “cap” which combines with a square to make a 3d shape. I tried to imagine what the shape formed by gluing the square and the curved shape would look like, but quickly reached the limits of my imagination.
Luckily, though, my wife was willing to help me sew a version.

It took two tries but eventually this shape emerged!

It is much flatter in reality than it was in my mind so seeing an actual version of the shape turned out to be really helpful.

I’m not sure what the next steps are for me. Either I have to get a better understanding of the Riemann mapping theorem (and I’ve already dug out my old complex analysis book for that) or maybe just play with some approximations and make some 3d prints like this one from Yoshiaki Araki that was part of a contest that Quanta Magazine had in their article:

The work trying to get a better understanding of these 3d shapes has been really fun. I’ll be really happy if I’m able to understand one or two more things from the DeMarco and Lindsey paper. It would be amazing to be able to make some (even very simple) shapes to show kids some new ideas from current math research.

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