Packing 16 Tetrahedra in Mathematica
This post was originally a Twitter thread.
Thanks to a suggestion from Ed Pegg, I explored the optimal packing of tetrahedra. (https://blog.wolfram.com/2010/08/30/tetrahedra-packing/)
Here is the best way to pack 16 tetrahedra: https://sketchfab.com/3d-models/tetrahedron-packing-original-7d331456fc0c4b6781fcad4df862d216
And here is a cleaned version where I’ve gotten rid of some of the coplanar faces so that the viewer can better see the interior structure: https://sketchfab.com/3d-models/tetrahedron-packing-cleaned-a8cc7cce9b9c42a398cf7236d006f3e7
The cleaning process itself was rather difficult.
It required finding the overlapping faces (Using Normal vectors!) and finding their boundary. But Mathematica didn’t allow me to do this for two-dimensional faces embedded in three dimensions! So I had to rotate the faces into a coordinate plane.
Then I could project into 2D, apply RegionUnion and extract the region boundary. After 3D-ifying the boundaries, they were re-rotated into space and thickened using the Tube command. They would be nice earrings…
Citation
@online{hanusa2022,
author = {Hanusa, Christopher R. H.},
title = {Packing 16 {Tetrahedra} in {Mathematica}},
date = {2022-10-05},
url = {https://hanusa.xyz/blog/posts/2022-10-05-packing-16-tetrahedra-in-mathematica/},
langid = {en}
}