# 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 ðŸ™‚

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.

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!

# Sharing “The Secret Life of Equations” with the boys

I saw Rich Cochrane’s The Secret Life of Equations at the book store yesterday and bought it for some fun with the boys. Here’s the book on Amazon:

I’ve done a few other projects previously in which the boys pick out ideas from a book. Here’s what they had to say flipping through this one.

My younger son like the chapter on the equation \$latex e^{\pi i} + 1 = 0:

By lucky coincidence Grant Sanderson has a neat new video on the subject – maybe that’ll be interesting to my son, too:

The idea that caught my older son’s attention was the heat equation. We haven’t quite gotten to calculus yet (ha ha ), but it was still fun to hear what he had to say:

I like the book even though it really isn’t intended for kids. We’ll probably do several more projects like this one over the next few months.