Lorenz attractor in 3 Dimension

This post is created by Khalilah Hamer of Math 401: Mathematics through 3D printing  at George Mason University.

This week we were challenged to look back at our previous assignments and redo or recreate one that did not have the best final project. I had a few slips here and there for projects but one print that I wanted to redo was my chaotic attractor print. I recreated the Aizawa attractor which I thought would be a really cool print and my finished project looked great on the computer, although with the need for natural filament making the print take much longer than originally planned, I had to scale down the print drastically. This caused the tubes of the attractor to become thin and brittle which was not what I had hoped for.

I am redoing the chaotic attractor and this time, to do more than just adjust the tube width, I decided to recreate another attractor. I recreated the Lorenz Attractor and this time making sure the tubes are greater than 1cm (the Aizawa was about 5mm). When making the tubes larger they sort blended together and it was harder to see the details so I decreased the 'timelength' to 30 to create a better visual, as 3D printing really helps the visual aspects.

To get more into the chaotic attractors they are defined as an attractor of chaotic dynamical systems in chaos theory. This is where two points on the attractor are near eachother at one time but will be far apart at later times as long as the system remains on the attractor. For example, this math can be used to “predict” or get a good guess at climate as it is chaotic and can’t be predicted. The Lorenz attractor is the set of solutions to the Lorenz system, founded by Lorenz himself. It was initially to resemble the convections of the atmosphere. It is said the shape resembles a butterfly and is sort of how the name the butterfly effect came to be as even when you know the simple initial conditions a small rush of wind from a butterfly’s wing can through the whole system in well, chaos. 

The Lorenz attractor is a strange attractor consisting of 3 equations as follows as well as the constants.

I used the initial conditions of 0 for x, y, and z. On my readings of the Lorenz attractor I got the constants as these are the most commonly used with this attractor.

An Ultimaker preview with half sphere stand

Recreating the mathematica code was more simple than the first time around as I could use my previous attractor code as a guide. Lastly I added a stand to help the print not collapse mid print and to allow the attractor to be displayed.

Final Project