A fun shape for kids to explore: the Permutohedron

I learned about permutohedrons from a comment by Allen Knutson on a prior blog post. See the first comment here:

A morning with the icosidodecahedron thanks to F3

I prnted the shape from Thingiverse and it was amazing!

“Permutahedron” by PFF000 on Thingiverse

We started the project today by examining the shape and comparing it to a few other shapes we printed. The comparison wasn’t planned – the other shapes just happened to still be on the table from prior projects . . . only at our house 🙂

Next we talked about permutations and the basic idea we were going to use to make the permutohedrons. We drew the 1 dimensional version on the whiteboard and talked about what we thought the 2 dimensional version would look like.

We used our zometool set to make a grid to make the 2 dimensional permutohedron. Lots of different mathematical ideas for kids in this part of the project -> coordinate geometry, permutations, and regular old 2d geometry!

Next we went back to talk about how PFF000’s shape was made. Here’s the description on Thingiverse in case I messed up the description in the video:

“The boundary and internal edges of a 3D permutahedron.

The 4! vertices are given by the permutations of [1, 3, 4.2, 7], with an edge connecting two vertices if they agree in two of the four coordinates. The 4D vertices live in a 3D hyperplane, namely the sum of the coordinates is 15.2.
Designed with OpenSCAD.”

This part of the project was a little longer, but worth the time as both the simple counting ideas on the shape and the combinatorial ideas in the connection rules are important ideas:

Finally we wrapped up by taking a 2nd look at the shape and also comparing it to Bathsheba Grossman’s “Hypercube B” which was also still laying around on our project table!

This was a really fun project that brought in many ideas from different areas of math. I’m grateful to Allen Knutson for the tip on this one!

One thought on “A fun shape for kids to explore: the Permutohedron

  1. You’re welcome! I’ll let you know when we put up the others in the family.
    Early taste: https://plus.google.com/+AllenKnutson/posts/SA79gzXQSL3

    I imagine it’s easy to make one of these with Zometools (just the external edges). It’s easy with Crazy Fort, e.g.

    Then the fun math project: how can you label the vertices with permutations, so that adjacent vertices only swap two adjacent numbers? (The internal edges in the 3-d printed one are for non-adjacent.)

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