This seems kind of cool, but also feels a little hackish. There are two pieces they are 'gluing together':<p>> <i>“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.”</i><p>And the cap comes from:<p>> <i>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.”</i><p>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?<p>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!<p>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 <i>other</i> 3D fractals which are more deserving of the name, IMO (e.g. the Mandelbulb: <a href="https://en.wikipedia.org/wiki/Mandelbulb" rel="nofollow">https://en.wikipedia.org/wiki/Mandelbulb</a>).<p>Then again, they're probably more interested in discovering deep properties about polynomials than making pretty things, unlike myself, so... go them.
I didn't why they got hung up, but I'm not a mathematician.<p>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.
there are some nice fast computations of Julia sets in Python
<a href="https://www.ibm.com/developerworks/community/blogs/jfp/entry/Fast_Computation_of_Julia_Set_in_Python?lang=en" rel="nofollow">https://www.ibm.com/developerworks/community/blogs/jfp/entry...</a><p>however one problem is always requires computing all the backwards iterates of the points. And there's <i>never</i> getting around that.<p>these two ladies offer a means of approximating the Julia set using a polygon. I have no heard of this.<p>there is a nice result by William Thurston they are using
"Shapes of polyhedra and triangulations of the sphere"
<a href="https://arxiv.org/abs/math/9801088" rel="nofollow">https://arxiv.org/abs/math/9801088</a><p>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.