Folding a dodecahedron into a cube has been one of my favorite projects to do with the boys. Our first few projects about a “dodecahedron folding into a cube” are here:

Today I had the boys work through the whole project on their own – just stopping every now and then to check in and hear about the progress.

Here are their initial thoughts after building the dodecahedron:

In the second part of the project the boys constructed one of the cubes that can be inscribed in a dodecahedron:

For the 3rd part of the project they “folded” the dodecahedron into a cube

Finally, the boys connected up the zome balls inside the cube and found an icosahedron.

Folding up the dodecahedron into a cube is one of my all time favorite math projects. It is such a surprise that the two shapes can be connected in this way, and it is really fun to explore this connection with our zometool set!

[had to write this in a hurry before the family headed off for a vacation – sorry that this post is likely a little sloppy]

Yesterday I gave a talk at a math camp for high school students at Williams College. The camp is run by Williams College math professor Allison Pacelli and has about 20 student.

The topic for my talk was the hypercube. In the 90 min talk, I hoped to share some amazing ideas I learned from Kelsey Houston-Edwards and Federico Ardila and then just see where things went.

A short list of background material for the talk (in roughly the order in the talk is):

(1) A discussion of how to count vertices, edges, faces, and etc in cubes of various dimensions

This is a project I did with my kids a few years ago, and I think helps break the ice a little bit for students who are (rightfully!) confused about what the 4th dimension might even mean:

(2) With that introduction I had the students build regular cubes out of the Zometool set I brought. Then I gave them some yellow struts and asked them to construct what they thought a hypercube might look like. From the prior discussion they knew how many points and lines to expect.

To my super happy surprise, the students built two different representations. I had my boys talk about the two different representations this morning. Funny enough, they had difference preferences for which was the “best” representation:

Here’s what my older son had to say:

Here’s what my younger son had to say:

At the end of this section of my talk I showed the students “Hypercube B” from Bathsheba Grossman (as well as my Zometool version):

(3) Now we moved on to looking at cubes in a different way -> standing on a corner rather than laying flat

I learned about this amazing way to view a cube from this amazing video from Kelsey Houston-Edwards. One of the many bits of incredible math in this video is the connection between Pascal’s triangle and cubes.

Here are the two projects I did with my kids a after seeing Houston-Edwards’s video:

After challenging the kids to think about what the “slices” of the 3- and 4-dimensional cubes standing on their corners would be, I showed them the 3D printed versions I prepared for the talk:

Part of my talk on hypercubes at Williams tomorrow -> 3D slices of a hypercube standing on a corner produce tetrahedrons and an octahedron pic.twitter.com/f7s8XM6kSB

I walked the students through how the vertices of a square correspond to the subsets of a 2-element set and then asked them to show how the vertices of a cube correspond to the subsets of a 3-element set.

There were a lot of oohs and ahhs as the students saw the elements of Pascal’s triangle emerge again.

Then I asked the students to find the correspondence between the 4-d cubes they’d made and subsets of a 4-elements set. I was incredibly happy to hear three different explanations from the students about how this correspondence worked – I actually wish these explanations were on video because I think Ardila would have absolutely loved to hear them.

(5) One last note

If you find all these properties of 4-D cubes as neat as I do, Jim Propp has a fantastic essay about 4 dimensional cubes:

By lucky coincidence, this essay was published as I was trying to think about how to structure my talk and was the final little push I needed to put all the ideas together.

Here are some thoughts from the boys on the figure and the proof. You can see from their comments that they understand some of the ideas, but not quite all of them.

Watching Mathologer’s video, I thought that the triangle proof about the irrationality of and the proof of the irrationality of using squares were something kids could grasp, but thought that the pentagon proof presented here was a bit more subtle. We may have to explore this one more carefully over the summer.

After discussing the proof a bit, I switched to something that I hoped was easier to understand. Here we talk about the different pairs of numbers that create fractions close to .

The boys were able to explain how to manipulate the pentagon diagram to produce the fraction 38/17 from the fraction 9/4 that we started with. From there the were able to also show that 161/72 was also a good approximation to :

Next we went to the computer to explore the numbers, and also to see how the same numbers appear in the continued fraction for .

In the last video we tried to do some of the continued fraction approximations in our head, but that wasn’t such a great idea. Here we finished the project by computing some of the fractions we found in the last video by hand.

I love Mathologer’s videos. It is amazing how many ways there are to use his videos with kids. Can’t wait to explore these “triangular squares” a bit more!

We’ve done a few projects on pyramids and tetrahedrons recently thanks to ideas from Alexander Bogomolny and Patrick Honner. Those projects are collected here:

One bit that remained open from the prior projects was sort of a visual curiosity. When you hold the zome Tetrahedron and zome Pyramid in your hand, it doesn’t look at all like the pyramid has twice the volume. Today’s project was an attempt to dive in a bit more into this puzzle.

We started by reviewing the ideas that Alexander Bogomolny and Patrick Honner shared:

Next we reviewed the geometric ideas that lead you to the fact that the volume of the square pyramid is double the volume of the tetrahedron.

Now we moved to the experiment phase – we put packing tape around the tetrahedron and the pyramid and filled them with water (as best we could). We then dumped that water into a bowl and used a scale to measure the amount of water. Our initial experiment led us to conclude that there was roughly 1.8 times as much water in the pyramid.

After that we repeated each of the measurements to get a total of 5 measurements of the volume of water in each of the shapes. Here are the results:

Definitely a fun project. I wish that we’d have gotten measurements that were closer to the correct volume relationship, but it is always nice to see that experiments don’t always match the theory!

My older son is on a school trip this weekend, so this project is just with my younger son (in 6th grade). I thought he’d had a lot of fun playing around with the program, so I let him explore it (with no instruction or even explanation) for about 10 min and then asked him what he thought was neat:

At the end of the last video he was playing around with the different numbers. I didn’t want to go into what those numbers represented, but I did think it would be great to hear some of his ideas and conjectures.

He found some ideas that seemed to work and a few that didn’t – so that was great to hear. By the end we’d found a shape that we could make from our Zometool set.

To finish this morning’s project we built the shape – here’s are his thoughts about having the shape in front of him vs seeing it on a computer screen:

This was a super fun project. I think it might be a nice challenge to try to dive a little deeper into the general Wythoff constructions that the Matt Zucker’s program is designed to explore. For now though, even with any details, the program is really fantastic for kids to play with.

[had to write this in more of a hurry than usual as 30 min of my morning was spent fishing for a dropped retainer that fell through a gap in our bathroom floor . . . . so sorry for the quite write up, but this project is a really fun way to get to hear a younger kid think about 3d geometry]

There were two really nice math ideas shard on twitter this week and I had no idea that they were related.

The first was a famous problem shared by Alexander Bogomolny:

Today – with just my younger son – I looked at a surprising connection between these two projects. We started by reviewing the Pyramid / Tetrahedron problem and then trying to guess the relationship between the volume of the two shapes.

Sorry that the lighting is so awful in these videos – unfortunately I only noticed after we were done.

Next I showed him the larger Tetrahedron with the inscribed octahedron. Although the main point of today’s project wasn’t Varignon’s theorem, I explained the theorem and asked my son to find some of the inscribed squares.

This connection was pointed out by Graeme McRae in this tweet:

Your talk led me to apply Varignon's Theorem to a skew (i.e. non-planar) quadrilateral, in particular to the vertices of a regular tetrahedron. It turns out Varignon's Theorem gives a quick way to show that a regular tetrahedron has a square cross section. https://t.co/HOynUFxpfB

At the end of the last video my son was starting to think about how volume scales. Since that’s an important point for this project I wanted to have all of those thoughts in one video.

It is interesting to hear how he tries to reconcile his mathematical thoughts about the volume of the two shapes with what he sees right in front of him.

Finally, we wrapped up by trying to find the relationship between the volume of the small tetrahedron and the volume of the pyramid.

I’m happy that my son is not convinced that the mathematical scaling arguments are correct. I can also say that holding these two objects in your hand it really does not look like the pyramid has twice the volume. Can’t wait to follow up on this.

It seemed like a great Zometool project, so this morning we cleared all the furniture out of the living room and gave it a go. My older son had something else going on today, so there were only two of us working on this project.

Here’s what my younger son thought of Perkins’s drawing:

Here are his thoughts after he completed the shape that he guessed would be the main building block for the project:

Here are his thoughts after we were just over half way done:

Finally, here’s are his thoughts on the completed project:

It is always fun to hear what kids have to say about shapes – and this project was a nice way to hear how my son’s thoughts about a fairly complicated shape evolved as we built the shape.

We’ve done a few other project like this one – this two old projects inspired by Kate Nowak and Anna Weltman comes to mind:

Patrick Honner was a terrific guest on the My Favorite Theorem blog today:

Hey, look, it's a new @myfavethm starring @MrHonner! Pause at the 4:18 mark to avoid spoilers (but turn it back on when you've thought about it for a little while) https://t.co/3vt5OBx03S

After listening to today’s My Favorite Theorem episode I wanted to do a follow up project – probably this weekend – but then I saw a really neat tweet just as I finished listening:

Your talk led me to apply Varignon's Theorem to a skew (i.e. non-planar) quadrilateral, in particular to the vertices of a regular tetrahedron. It turns out Varignon's Theorem gives a quick way to show that a regular tetrahedron has a square cross section. https://t.co/HOynUFxpfB

Well . . . I had to build that from our Zometool set and ended up finding a fun surprise, too. I shared the surprise shape with my older son tonight and here’s what he had to say:

What a fun day! If you are interested in a terrific (and light!) podcast about math – definitely subscribe to My Favorite Theorem.

I’m definitely enjoying the talks, but also wondering if there are ideas – even small ones – that you can take from the talks and share with kids. My hope is that kids will enjoy seeing ideas and concepts that are interesting to mathematicians.

Today I tried out an idea from Federico Ardila’s talk with my younger son (who is in 6th grade). The idea related to an interesting shape called the “permutahedron.”

We began with a quick explanation of the idea and looked at some simple cases:

Next we moved to building the permutahedron that comes from the set {1,2,3}. At the end of the last video, my son speculated this shape would have some interesting symmetry. We used our Zometool set to build it.

One thing I’m very happy about with this part of the project is that building this permutahedron is a nice introductory exercise with 3d coordinates for kids.

Finally, we talked about the permutahedron that comes from the set {1,2,3,4}. My son had some interesting thoughts about what this shape might look like. Then I handed him a 3d printed version of the shape and he had some fun things to say 🙂

Definitely a fun project for kids, I think. Making the hexagon was fun and also a nice little geometric surprise. Exploring the 3d printed shape was also really exciting – it is always great to hear what kids have to say about shapes that they’ve never encountered before.