Michael Gagliardo

Professor of Mathematics
California Lutheran University
Thousand Oaks, California, USA

I hesitate to talk about chaos after all of the events in 2020, but, hopefully we can find beauty in mathematical chaos. These two 3D prints are approximations of the Aizawa Attractor and the Dequan Li Attractor. Many people are familiar with the `butterfly shape" of the Lorenz Attractor, however, these two lesser known attractors are just stunning to me.

The method for creating these prints is described in the paper, "3D Printing Chaos", which was presented at the Bridges conference in 2018. I used Newton's method to create a solution in Mathematica followed by a python script to import the solution into Rhino as a NURBS curve. This method is generalizable to any dynamical system and the code is available through the github link.

The Aizawa Attractor
8 x 10 x 10 cm
3D Print - Plastic

This 3D print of the Aizawa Attractor was inspired by the computer representations found at chaoticatmospheres.com. I love how you can follow the flow around the sphere and back up through the twisting center in a forever non-repeating loop. According to a recently published article in the AMS Notices, the correct name for the equations that give rise to this attractor is the Langford equations and not the Aizawa equations. The print itself is very flexible. The center spiral is loose and not entirely fused which gives an almost springy feel to the piece when handled. In my 2018 Bridges conference paper, I discuss bifurcation points of the equation where the solutions demonstrate chaotic, periodic, quasi-periodic and convergent behavior.

The Dequan Li Attractor
10 x 8 x 6 cm
3D Print - Plastic

This is a 3D Print of the Dequan Li Attractor, also inspired by the work at chaoticatmospheres.com. This solution looks like a potter's wheel gone wrong. I am fascinated how dynamic the 3D print looks despite the static medium. The curves of this solution that are separate from the rest main body of the print are flexible and bounce when handled.