Modeling a Tesseract

Consider a 3D printed cube. This cube will cast a different shadow onto a piece of paper when the light source moves around.

series of images showing the different shadows cast by a cube when a single light source is moved around.

Beings living in a 2D world will experience the cube differently depending on how the shadow is cast onto their world.

Similarly, a 4D cube, or Tesseract, can also cast a shadow onto our 3D world. It is challenging to think about this because we do not directly experience the world in 4 dimensions. Nevertheless, I was able to model a tesseract using Python and Rhino. Specifically, I modeled a 4D tesseract and its perspective projection onto 3D space. This model will change as the tesseract rotates in 4D space. The projections were modeled in Rhino using the RhinoCommon SDK.

I can build the tesseract and rotate it around any plane in 4D space. The whole process is automated so I can bulk generate Rhino models using Python. Here is an animation the tesseract rotating around the X-Z plane:

3D Printing

Next I need to 3D print several versions of the tesseract as it rotates around the X-Z plane. I need to think carefully about the sizing of the print. There is a minimum edge width that I can reliably print with a 3D printer, but I also don't want the widest edges to inordinately increase the time it takes to print. I need to adjust the scaling and the perspective projection to give me a reasonable set of prints. After that, I will need some patience and a little bit of luck that the 3D printers will behave as I need them to.