Visualizing the 5d permutohedron with kids

Last night as part of a linear algebra project I was doing with my older son, we found out that you can orient a 4d permutohedron in 3 space so that all of the vertices have integer coordinates:

Today I wanted to explore that idea a bit more and also include my younger son. So, I thought it would be fun to see if we could find a way to see what the 5d permutohedron looks like by looking at slices of it in 4d.

I started by reviewing the 3d permutohedron and how it is embedded in 2 dimensions. It was nice to go back to the beginning here – especially so that we could explore how slicing with lower dimensional slices works.

Next we tried the same “visualization by slicing” idea with our 4d permutohedron embedded in 3 dimensions:

Finally, and sorry this one is long, we got to the heart of today’s project. Here we’ll be using some code I wrote in Mathematica to view 3d slices of the 5d permutohedron emedded in 4d space. It is close to a miracle that I was able to get these visualizations to work correctly – maybe the extra hour this morning helped! It was super fun to hear the boys talk about what they saw with these shapes: