Playing with “zonish polyhedra” and the icosidodecahedron

Yesterday I stumbled on George Hart’s website and found some neat ideas to play around with using our Zometool set:

https://www.georgehart.com/virtual-polyhedra/zometool.html

https://www.georgehart.com/virtual-polyhedra/zonish_polyhedra.html

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!



Talking 4d shapes with my kids

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.

Having kids play with 4 and 5 dimensonal Hasse diagrams

Yesterday we did a fun project on Hasse diagrams:

https://mikesmathpage.wordpress.com/2020/05/24/exploring-hasse-diagrams-with-kids-thanks-to-martin-weissmans-an-illustrated-theory-of-numbers/

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:

So for a challenge project this morning I had the boys try to figure out how a Hasse diagram would work in 4 dimensions and in 5 dimensions.

Here’s how the 4d case went:

The 5d case was significantly more challenging – but they got there! Here’s the explanation of their work:



Who would have ever thought that a 5d cube appearing in your twitter feed would be exactly the thing you needed to see for a new math project!

Revisiting a dodecahedron folding into a cube

Back in 2016 Simon Greg showed me an incredible GIF of a dodecahedron folding into a cube:

dodecahedron fold

which he found on this other amazing blog post by Herman Serras:

The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron

Today we decided to revisit our old project of making the shapes from our Zometool set. We started by looking at the gif:

Next I had the boys play around with the zometool set to try to make this shape. They worked for 15 min and made these shapes:

Here’s a closer look at the dodecahedron and the folded shape:

Finally, we took a look at one last surprise:

Talking about the 2d projection of a 5d cube with my younger son

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

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!

A fun zometool project with decagons

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

At the time I thought it would be really fun to see if we could make these shapes with our Zometool set. We tried it out this morning – and success!

Screen Shot 2020-02-01 at 10.20.28 AM

After we finished building them, I had my younger son talk about the shapes. Here’s the discussion of the first three:

And here’s the description of the next 4:

I’m always super happy when shapes I see on twitter can be made from our Zometool set!

We are going to try to make the shape corresponding to the projection of the 5 dimensional cube for tomorrow’s project. Should be really fun.

Finding the coordinates of the vertices of a Tetrahedron and an Octahedron

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:

A Zometool Icosahedron project inspired by Steve Phelps

I saw a neat tweet from Steve Phelps yesterday:

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:

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:

A Morning with the 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.