George Mason University Math 401: Mathematics Through 3D Printing

Nick Maranto 11/21/24

This week's project was about chaotic attractors. A chaotic attractor is a dynamical system that has an infinite number of periodic orbits. Also, when you make a small change in the initial condition in a chaotic attractor leads to large differences in the trajectory. When you take a cross section of a chaotic attractor, it is comprised of a Cantor set. In this project, I printed an attractor with two different parameter values.

The two 3D prints are Thomas attractors. A Thomas attractor is a dynamical system with the differential equations:

x’ = sin(y) – b*x

y’ = sin(z) – b*y

z’ = sin(x) – b*z.

The constant b is varied between the two prints. The print with chaotic trajectory has a b value at 0.2081 while the periodic trajectory has a b value at 0.19. Visually, the two prints are drastically different with such a small change in the parameter. This dynamical system models damping force.

The attractor shown was printed on a UltiMaker S7. To create the print, we used PLA and dissolvable filament. The PLA was used for the attractor while the dissolvable filament was used for the supports. Once it was done printing, the attractor was left in water until all the dissolvable filament was off the attractor. The approximate time it took to print was three to four hours. To display both prints, a base plate can be printed. To do this, build a plate and carve out a sphere for the print to rest in.

Attached are 3D renders from Mathematica, a photo of the attractor with a build plate, the STL files for both prints, and the Mathematica code that calculates and graphs the trajectory of the dynamical system. The photos show different angles of each print. In the Mathematica code, different trajectories can be graphed by changing the b value.

Citations: https://arxiv.org/html/2408.09525v1 Lucas, S. K., Sander, E., & Taalman, L. (2020). Modeling Dynamical Systems for 3D printing. Notices of the American Mathematical Society, 67(11), 1. https://doi.org/10.1090/noti2187