# 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:

# Labeling each vertex of a permutahedron is a terrific mathematical exercise for kids

Yesterday we did a fun project exploring a permutohedron:

Last night I thought it might be neat to have the kids try to label the vertices of a permutohedron with the permutations represented by each vertex. Fortunately, it was possible to build a truncated octahedron with the green Zometool struts:

We started out today’s project by talking about the rules for making a permutohedron in different dimensions. Here I used the labeling of the permutation of 3 objects as a base case to make sure the boys understood the directions properly.

Next I had the boys label each vertex of the permutahedron with the permutation of {1,2,3,4} that the vertex represented. Then, they talked about the process of figuring out the right labels.

I’m sorry that the video below runs 10 min, but if you listen to the whole discussion I think you’ll see that seemingly straightforward act of labeling these vertexes is a terrific mathematical exercise for kids.

# A morning with the permutohedron

Today we are revisiting an old project on a really neat shape -> the permutohedron:

“A fun shape for kids to explore – the permutohedron

I learned about this shape thanks to Allen Knutson at Cornell – he included a fun pic of a large permutohedron in the comment of the blog post above:

He also pointed me to a 3d print on Thingiverse that we used in the last project and again today:

“Permutahedron” by PFF000 on Thingiverse

So, I started today by having the boys describe the 3d printed shape. We have two versions – a larger one that unfortunately broke a little and a smaller – but in one piece! – version. Here’s what the boys had to say about the shapes:

Next I had the boys read the Wikipedia page on the permutohedron for about 10 min and then we discussed some of the ideas that they thought were interesting:

Finally, we built the 2-D permutohedron and showed how it was embedded in a 3d grid:

Definitely a fun project and it is always great to be able to have kids hold interesting math ideas in their hands!

# A neat project with a dodecahedron

Saw a really neat tweet this morning:

I thought it would be fun to see what the boys thought of this shape and then try to building using our Zometool set.

First I showed them the video:

Next we spent 20 min building the outside shell of the shape, but for now left the inside mostly empty. Here’s what the boys thought of the shape:

Finally, here is the completed shape – it is a nice little miracle that we could make the whole thing with the Zometool set!

Such a fun project! Happy for the lucky break from twitter this morning 🙂

# A bonus project on a Zometool icosahedron

We’ve done two projects on platonic solids recently:

Talking about Angles in Platonic Solids

Following up on our angles in platonic solids project

In the last project my younger son explored two different kinds of “golden rectangles” inside of the icosahedron. I thought it would be fun to try to fill in the entire shape with the rectangles, so today the boys took on that challenge.

Here’s their discussion of the shape made by filling in all of the large golden rectangles in the icosahedron:

Next we turned to the shape made by filling in the smaller golden rectangles. These were a little harder to make. Since the first shape took a bit longer to make than we expected, we only filled in 10 of these rectangles and avoided the problem of dealing with ones that overlapped.

To wrap up we removed the struts from the original icosahedron to get a better view of the shape formed by the rectangles:

Definitely a fun project. As always, it is incredible how easy (and fun!) it is to explore 3d shapes with a Zometool set.

# Following up on our “angles in Platonic solids” project

Yesterday we did a fun project on angles in Platonic solids:

Talking about Angles in Platonic Solids

We ended up getting a really neat comment from Allen Knutson on that project. He said:

“You should look for the three orthogonal golden rectangles in an icosahedron! They’re easy to see in a Skwish toy.”

My older son was working on a different math project today, so I had my younger son build an icosahedron out of zome and look for those rectangles. Here’s what he had to say after building the shape:

During his description he found a second rectangle. So, off camera, he filled in that rectangle and then had a bit more to say:

So, thanks to Allen Knutson for the comment that inspired this project, and thanks (as always!) to Zometool for making it so easy to get kids talking about math!

# Talking about angles in Platonic solids

My younger son wanted to do a Zometool project today and since my older son is currently learning about the dot product, I thought it would be fun to talk about angles in some platonic solids.

This idea turned out to be one that was better in my mind than it was in practice – ha! – but it was still a nice project even though it got a bit messy.

We started by talking about angles in a cube:

Next we moved to the octahedron:

Here we go through the steps to calculate the angle between two faces in the octahedron:

Finally, we wrap up by looking at the fun surprise that a hypercube has a 30-60-90 triangle hiding in it! My younger son got a little confused about how to find the lengths of some of the vectors we were looking at, so we went slow. It is really fun to see how some relatively simple ideas let you explore hard to visualize objects like a 4-dimensional cube!

# Exploring an amazing tweet from John Carlos Baez with my younger son

I saw an incredible tweet from John Carlos Baez last week:

Here’s the picture in case the tweet isn’t embedding all that well for you:

I thought exploring some of these shapes would make a great project for kids, so I began by asking my son (in 7th grade) for his thoughts on the shapes he was seeing:

Next we built a few of the cubes from our Zometool set and talked about some of the shapes. First, though, I asked my son to give his definition of what an n-dimensional cube was:

Finally, we played around by a different version of a 4-dimensional cube – “Hypercube B” by Bathsheba Grossman. This amazing version of the hypercube makes amazing shadows and my son was able to find a projection that was a little closer to the projection of the 4d cube in Baez’s tweet:

Also, here’s a video I made a while back showing some other (almost freaky) 2d projections from our Zometool model of Hypercube B:

Definitely a fun project – thanks to John Carlos Baez for sharing some of his ideas about higher dimensional cubes on twitter!

# Finding the volume of a rhombic dodecahedron with our zometool set

Yesterday we did a neat project inspired by a tweet from Alex Kontorovich:

Sharing a 3d geometry idea from Alex Kontorovich with kids via zometool

At the end of that project a question about finding the volume of a rhombic dodecahedron came up. Since I was going to be out this morning (and my older son was working on a calculus project) I asked my younger son to play around with the Zometool set and see if he could actually find the volume.

Fortunately he was able to – here’s how he described his work:

# Sharing a 3d geometry idea from Alex Kontorovich with kids via Zometool

I saw an interesting tweet from Alex Kontorovich earlier this week:

We’ve looked at but the Cuboctahedron and the Rhombic dodecahedron before, but I thought it would be fun to revisit the shapes. I also hoped that we’d be able to recreate the shape in the picture with our Zometool set.

So, first we built a cuboctahedron and the boys talked about what they saw in the shape:

At the end of the last video the boys thought that the dual of the cuboctahedron would possibly also be another cuboctahedron. Off camera we built the dual, and happily were able to recreate the shape from Kontorovich’s shape!

They were a little worried that we didn’t have the “true” dual, but I think they came around to believing that these two shapes were indeed duals:

Definitely a fun project – it is always fun to see what you can make with a Zometool set. Maybe tomorrow we’ll revisit an old project of finding the volume of a rhombic dodecahedron. That’s another project which Zometool really brings a lot to the table.