We did a project talking about it back then, but I decided to revisit it today with my younger son.

First we recreated part of the drawing using our Zometool set, and then I had my son share his thoughts about the shapes – it is always so fun to hear what kids see when they look at shapes:

Next I had my son talk about the Zometool shape we made:

Today we explored the and idea that Ardila discussed in the podcast – finding paths on the permutahedron.

We started by just reviewing what the shape is and what it represents:

Next we tried an easy example of finding a path on the permutahedron going from a random permutation back to the correct order:

For the last part of the project we tried a more complicated scramble of the cards and found that walking back to the unscrambled state would take a minimum of 3 steps:

I love playing with this shape with kids. It is a great way to get them talking about fairly advanced mathematical ideas and also allows them to see a really neat 3d shape that research mathematicians find interesting!

After seeing these pages my younger son and I built one of the models and talked about it:

Today we explored the shape a bit more by building an icosidodecahedron and comparing it to the shape from yesterday:

Two wrap up today we looked at how spherical the icosidodecahedron is. I would have like to do the same exercise for the “zonish polyhedra” we were look at, but I’m not sure how to calculate the volume of that shape.

This was a really fun project – it is absolutely amazing how easy it is to explore 3d geometry with a Zometool set!

I don’t know why, but this zometool shape we made a few years ago based on Bathsheba Grossman’s Hypercube B migrated back down to the living room this week:

Seeing how that zome creation seems to change as you walk by it once again this week made me want to do a project revisiting 4d shapes with the boys.

We started by looking at a few shapes that we’ve played with before:

Next we looked carefully at Hypercube B by Bathsheba Grossman:

Now I had the boys watch the video about the Zometool version of Hypercube B and react to it:

Next we went to the Wikipedia page for the hypercube and look at some of the 2d representations. The boys reacted to some of the pictures and I asked them to pick one and draw it.

Here are their drawings and explanations. One fun surprise is that after they finished their drawings they noticed that they chose the same shape!

This was a fun project and not meant to dive into great detail. I’m happy that the boys are getting comfortable thinking about higher dimensions – it has been really fun to explore ideas from higher dimensional geometry with them.

The boys got the hang of a few relatively simple examples but also noticed that going to numbers with 4 prime factors would get pretty hard to draw.

After we finished the project I saw a post on twitter about a 5d cube and was reminded that we had a 2d projection of a 5d cube hanging on our living room wall:

We have the projection of the 5d cube you can make with zometool hanging on our living room wall π pic.twitter.com/yTVJoJFhjc

Last week I saw this really neat tweet from Tom Ruen:

The projections are closely related to quasicrystals as well. The Penrose tilings come from projections of a 5-cube (which stack infinitely in 5D like cubes in 3D). This shows all the rhombic dissections of a decagon.https://t.co/hmk3Vt0fHnhttps://t.co/knff5ywvn9pic.twitter.com/PRClCio2Hu

Yesterday my younger son and I talked through the decagons after building them from our Zometool set. Today we talked about the projection of the 5d cube.

Here are his initial thoughts:

My son was interested in comparing this 5d cube shape to a shape that we’d built previously. So we got that shape and continued the comparison. We also talked a bit about where else the number 5 appeared in the 5d cube and in our shape:

I’m so happy to have seen the conversation that Nalini Joshi got started on Twitter last week. We’ve had two super fun projects so far inspired by it!

Earlier in the week I saw a really neat twitter thread that had this post:

The projections are closely related to quasicrystals as well. The Penrose tilings come from projections of a 5-cube (which stack infinitely in 5D like cubes in 3D). This shows all the rhombic dissections of a decagon.https://t.co/hmk3Vt0fHnhttps://t.co/knff5ywvn9pic.twitter.com/PRClCio2Hu

My younger son is starting to learn about coordinates in 3 dimensions. I thought that spending a little time finding the coordinates of the corners of a tetrahedron and an octahedron would make for a nice project this morning.

We started with the tetrahedron and found the coordinates for the bottom face. Once nice thing about the discussion here was talking about the various choices we had for how to look at the tetrahedron:

Having found the coordinates for the bottom face, we now moved on to finding the coordinates for the top vertex:

Now we moved on to trying to find the coordinates for the corners of the octahedron. Here the choices for how to orient the object are a little more difficult:

Finally, we talked through how we would find the coordinates of the octahedron if we had it oriented in a different way. This was a good discussion, but was also something that confused the boys a bit more than I thought. We spent about 10 min after the project talking through how to find the height. Hopefully the discussion here helps show why this problem is a pretty difficult one for kids:

It looked like it could make a neat project both on the computer and with our Zometool set.

First I had my younger son look at Phelps’s visualization – one really interesting observation he had was that the intersecting lines inside the icosahedron dodecahedron:

Nest I had my older son look at a similar program in Wolfram’s Demonstration Project. The thing that caught his attention was all of the underlying structure:

We also created a zometool version of the icosahedron with all of the diagonals. We tried to see if we could see the same interesting things that we saw in the computer programs using the Zome shape:

Later in the day we did build a slightly larger icosahedron in which the diagonals did intersect on a zome ball. This allows you to see the dodecahedron that my younger son thought was there: