Julia's Coffee Table
Lucas Newman
George Mason University - Math 401 - Mathematics Through 3D Printing
October 25, 2021

This math art object is a blueprint for a coffee table with a beautiful, branching, fractal tabletop. It's certainly an odd looking surface, with thin and irregular spindles, but if it was constructed to be large enough there would be no actual difficulty placing mugs and books on the arteries. I believe it could occupy a lot of living rooms in a unique and visually cutting way.

The shape of the table is created in reference to something called a Julia set. The concept of a set being "Julia" comes from function iteration. A function formally "relates" the elements of one set to the elements of another set - for most functions most of the time it is appropriate and fun to imagine that when you apply a function to an object you are pushing that object through a special tunnel that transmogrifies it into something else. When we "iterate" a function, we apply it in succession, so that the image of the previous usage becomes the argument of the next one. It's as if we make a tunnel, but this time it is a very long tunnel where each piece of pipe is made from the original function.

When functions are applied in this iterative way to a single element an indefinite number of times one of two things happen: 1) the iteration converges to a stable point, which is then the limit of the iteration, or 2) the iteration diverges, either oscillating indefinitely or soaring off to an unobtainable extreme. By paying attention to this binary outcome, it is possible to classify the elements of a set with respect to a given function F with this question - "When I send this element down an infinite tunnel created by successive iterations of F, does the resulting computation produce a stable point, or does the sequence of iterations diverge?" Skipping over some analytical details, we can explain the Julia set with respect to the function F as that set of points which converge to a stable outcome during this kind of iteration.

The surface in this object is specifically the Julia set of the function: f(z) = z^2 − 0.8i, with z as any element of the complex plane mapped to coordinates in R2. Functions of the form f(z) = z^2 + c are popular targets for Julia visualizations because they have a behind the scenes connection to the Mandelbrot set - small changes to the parameter c result in a wide variety of beautiful and easy to discover graphic art images. My Julia set data was created in Mathematica, mounted onto the object depth with the rich 3D plotting tools available in that platform, and then exported to a .stl.

Citations -

https://www.marksmath.org/visualization/julia_sets/
https://archive.bridgesmathart.org/2010/bridges2010-247.pdf