Thomas Attractor: Parameters 0.18 and 0.205

Kaitlyn Sullivan
11/21/2024
George Mason University Math 401: Mathematics Through 3D Printing.

A chaotic attractor is characterized by its sensitivity to slight changes in initial conditions and parameters, creating vastly different and seemingly unpredictable “chaotic” behavior. Chaotic attractors can arise from several different systems of equations. Most famously is the Lorenz attractor resembling a butterfly.

This attractor is a 3D model of the Thomas chaotic attractor. Modeling an attractor is done by tracking several points over a certain time length and modeling the path they travel. By doing this with many points, the shape of the Thomas attractor comes into view.

This Thomas attractor is modeled using initial conditions of x0 = 1.1, y0 = 1.1 and z0 = -0.1 with the system of equations for a Thomas attractor as seen below:
F1[x, y, z] := Sin[y] - b x
F2[x, y, z] := Sin[z] - b y
F3[x, y, z_] := Sin[x] - b z
The focus of this project was to analyze how chaotic attractors act under different parameters. Using parameters b = 0.205 and b = 0.18, we see two very different objects. Object 1 is half of Object 2, but still resembles the chaotic behavior of the Thomas attractor.

Each object has been scaled up to 400% in Ultimaker Cura from its initial export from Mathematica. For Object 1 (b = 0.205) and Object 2 (b = 0.18), their corresponding stands have already been scaled up to fit Object 1 and 2. The objects were printed using dual extruders, where one extruder was dissolvable filament used for the supports and the other was regular filament for the object. The dissolvable filament was critical in ensuring these objects could be printed with enough detail and thin parts. Both were printed using Extra Fine resolution on an UltiMaker S5. The stands were printed with PLA on a Creality Ender-3 v2 Neo with Normal resolution and 10% infill and triangle supports. Object 1 takes about 3 hours and Object 2 takes around 10. The stands printed together take around 2.5.

It is recommended that you follow the specific instructions for whatever filament you are using. Generally, dissolvable filament is PVA (polyvinyl alcohol) where the recommended nozzle temperature is between 180 - 210°C and bed temperature is between 45 - 60°C. The recommended conditions for PLA filament is nozzle temperature between 190°C and 220°C and bed temperature between 50°C and 60°C.

I highly recommend viewing some other interesting chaotic attractors on https://www.dynamicmath.xyz/strange-attractors/. The site provides great visuals of the behavior of these attractors. To use the Mathematica code attached, simply modify the system of equations, initial conditions, parameters, and timelengtha and timelengthb as needed to create your 3D model of a chaotic attractor.