## Some beautiful geometry in a challenge problem from Alexander Bogonolny

I did the project below with the boys on Sunday before they went off to camp for a week. The idea wasn’t to get into heavy math, but rather just a relaxed walk through some fun shapes. We got one detail wrong in the 4th video which I was sort of kicking myself for, but then I saw a tweet from Nassim Taleb showing some of the geometry in a different problem that Alexander Bogonolny had posted and it made me realize the connection between the algebra and geometry in our problem was still fun to show:

So, despite the error I thought I would publish the project anyway.

Here’s the original problem:

Below are the videos showing our walk through the geometry. First, though, here’s the quick introduction to the problem:

After that intro we looked at the region described by the constraint in the problem. We have to thicken up the region a little bit using the absolute value function in order to see it, so the Mathematica code looks a bit more complicated than in the problem, but that extra complexity is just to make the picture easier to see.

One cool thing about our discussion here is that my younger son thought there should be 3 fold symmetry in the shape because there was 3 fold symmetry in the equation 🙂

Now we looked at the situation in which the surface achieves the maximum value subject to the constraint in the problem. My younger son made the nice observation that the two surfaces appeared to be “blending together” at certain points. That “blending” is an important idea in Lagrange Multipliers – though, don’t worry, we aren’t going down that path today.

Next we looked at the minimum value of the surface subject to the constraint in the problem. The error I made here was accidentally reversing the two surfaces. The fixed surface – the one describing the constraint – is now on the outside rather than the inside.

Finally, I asked the kids to pick a value smaller than 45/4 for the curve so that we could see what happened. Unfortunately they picked 7 which is too small – there’s no surface! – so they chose 10 and that allowed us to see that the shrinking surface inside of the original shape. Also we can see fairly clearly (after some rotation) that the two shapes do not intersect.

Definitely a fun project showing the boys a beautiful side of a really challenging problem.

## Working with the PCMI books part 2: coloring an octahedron

Last week we got the PCMI books:

Our first project involved a neat problem about understanding the number 0.002002… in different bases:

Playing around with the PCMI books

Today I was looking for another fun problem and found another problem that I thought would make a fun project:

Barbara has an octahedron, and she wants to color its vertices with two different colors. How many different colorings are possible? By “different” we mean that you can’t make one look like the other throu a re-orientation.

I started by introducing the problem and asking the kids what their initial ideas were:

They had a couple of pretty good ideas including some basic ideas about symmetry. Using those ideas we began counting the different colorings:

We counted the cases in which 3 vertices were black and 3 vertices were red. This case proved to be tricky, but going through it slowly got us to the correct answer.

Finally, as a fun little extension, I asked them to find the number of ways to color the faces of a cube with two colors. Having solved the octahedron problem already, this one went pretty quickly, and they even noticed the connection between the two problems 🙂

I like this problem. I’m glad that the boys were able to see some of the basic ideas. When you add more colors the counting gets much more difficult and some pretty advanced math comes into play. The number of colorings with “n” colors is:

$(n^6 + 3n^4 + 12n^3 + 8n^2) / 24$

The different terms correspond to different symmetries of the cube / octahedron. We’ll have to wait a few more years to cover the complete details 🙂

## Working through “Euler’s Gem” with kids

A few weeks ago I stumbled on Diave Richeson’s book Euler’s Gem:

Although the book is not intended for kids, it is written for a general audience and I thought the boys might enjoy working through the book slowly. I’ve been having them read one chapter per day and they are really having fun with the ideas.

Today I asked each of them to talk about what they’ve learned through the first 5 chapters. Here’s what my younger son had to say:

Here’s what my older son had to say:

It is fun to hear what they are taking away from this book and also really nice to hear that both of them really do like the book. I’ve not tried an experiment like this one before, but the book is so well written that I really do think that with a little bit of help here and there kids can understand most of it.

## One my time through F – E + V = 2

We did a fun project earlier in the week inspired by Dave Richeson’s book:

That project is here:

Looking at Dave Richeson’s “Euler’s Gem” book with kids

During the project the kids had a little trouble counting the verticies, edges, and faces of one of the complex shapes. We solved the problem with our Zometool set, but I wanted to try a different approach and printed the shapes again:

So, with these shapes I went through the project again. First a quick review:

Next, now that we have shapes that fit together, can we count the faces, verticies, and edges?

My younger son was still having a little bit of trouble seeing the number of edges, so we slowed down a bit:

Finally we did a quick recap of how the cube helped us. I was trying to get the boys to think about the shape without touching it, but wasn’t super successful.

This was a fun 2nd look at the F – E + V = 2 formula. We’ll be doing more projects based on Richeson’s book throughout the summer.

## Looking at Dave Richeson’s “Euler’s Gem” book with kids

I stumbled on this book at Barnes & Noble last week:

It is such a delightful read that I thought the kids might enjoy it, too, so I had them read the introduction (~10 pages).

Here’s what they learned:

Next we tried to calculate Euler’s formula for two simple shapes – a tetrahedron and a cube:

After that introduction we moved on to some slightly more complicated shapes – an icosahedron and a rhombic dodecahedron. The rhombic dodecahedron gave the kids a tiny bit of trouble since it doesn’t have quite the same set of symmetries as any of the Platonic solids:

Now we tried two very difficult shapes:

We studied these shapes last week in a couple of projects inspired by an Alexander Bogomolny tweet:

Working through an Alexander Bogomolny probability problem with kids

Connecting yesterday’s probability project with a few old 3d geometry projects

I suspected that this part would be difficult, so I had them count the faces, edges, and verticies of the two shapes off camera. Here’s what they found:

So, since the boys couldn’t agree on the number of verticies, edges, and faces of one of the shapes, I had them build it using our Zometool set to see what was going on. The Zometool set helped, thankfully. Here’s what they found after building the shape (and we got a little help from one of our cats):

Definitely a fun project. It was especially cool to hear the kids realize that the shape they were having difficulty with was (somehow) a torus. Or, as my older son said: “In the torus class of shapes.” I’m excited to try to turn a few other ideas from Richeson’s book into projects for kids.

## Connecting yesterday’s probability project with a few old 3d geometry projects

In yesterday’s project we were studying a fun probability question posed by Alexander Bogomolny:

That project is here:

Working through an Alexander Bogomolny probability problem with kids

While writing up the project, I noticed that I had misunderstood one of the
geometry ideas that my older son had mentioned. That was a shame because his idea was actually much better than the one I heard, and it connected to several projects that we’ve done in the past:

Learning 3d geometry with Paula Beardell Krieg’s Pyrmaids

Revisiting an old James Tanton / James Key Pyramid project

Overnight I printed the pieces we needed to explore my son’s approach to solving the problem and we revisited the problem again this morning. You’ll need to go to yesterday’s project to see what leads up to the point where we start, but the short story is that we are trying to find the volume of one piece of a shape that looks like a cube with a hole in it (I briefly show the two relevant shapes at the end of the video below):

Next we used my son’s division of the shape to find the volume. The calculation is easier (and more natural geometrically, I think) than what we did yesterday.

It is always really fun to have prior projects connect with a current one. It is also pretty amazing to find yet another project where these little pyramids show up!

## Working through an Alexander Bogomolny probability problem with kids

Earlier in the week I saw Alexander Bogomolny post a neat probability problem:

There are many ways to solve this problem, but when I saw the 3d shapes associated with it I thought it would make for a fun geometry problem with the boys. So, I printed the shapes overnight and we used them to work through the problem this morning.

Here’s the introduction to the problem. This step was important to make sure that the kids understood what the problem was asking. Although the problem is accessible to kids (I think) once they see the shapes, the language of the problem is harder for them to understand. But, with a bit of guidance that difficulty can be overcome:

With the introduction out of the way we dove into thinking about the shape. Before showing the two 3d prints, I asked them what they thought the shape would look like. It was challenging for them to describe (not surprisingly).

They had some interesting comments when they saw the shape, including that the shape reminded them of a version of a 4d cube!

Next we took a little time off camera to build the two shapes out of our Zometool set. Building the shapes was an interesting challenge for the kids since it wasn’t obvious to them what the diagonal line segments should be. With a little trial and error they found that the diagonal line segments were yellow struts.

Here’s their description of the build and what they learned. After building the shapes they decided that calculating the volume of the compliment would likely be easier.

Sorry that this video is a little fuzzy.

Having decided to look at the compliment to find the volume, we took a look at one of the pieces of the compliment on Mathematica to be sure that we understood the shape. They were able to see pretty quickly that the shape had some interesting structure. We used that structure in the next video to finish off the problem:

Finally, we worked through the calculation to find that the volume of the compliment was 7/27 units. Thus, the volume of the original shape is 20 / 27.

As I watched the videos again this morning I realized that my older son mentioned a second way to find the volume of the compliment and I misunderstood what he was saying. We’ll revisit this project tomorrow to find the volume the way he suggested.

I really enjoyed this project. It is fun to take challenging problems and find ways to make them accessible to kids. Also, geometric probability is an incredibly fun topic all by itself!

## Revisiting Kelsey Houston-Edwards’s Hypercube video

Last week Kelsey Houston-Edwards published a fantastic video about hypercubes – it is one of the best math videos I’ve ever seen:

Here’s our project on her video:

Kelsey Houston-Edwards’s hypercube video is incredible!

Today while the kids were at school I wrote a couple of Mathematica functions to replicate her results. Writing the code to make these shapes is actually a pretty fun exercise in linear algebra and trig, but that’s a little more than I felt like sharing with the kids just now 🙂

Instead I had them look at the shapes on the screen and tell me what they thought. The first video in each with each kid shows the shape made by a plane intersecting a 3d cube standing on its corners. The second video shows the 3d intersection of a hyperplane perpendicular to the long diagonal of a 4d cube intersecting that cube.

Here’s what my older son had to say:

(a) The 3d cube being cut by a (slightly thick) plane:

(b) The 4d cube being cut by the hyperplane:

Here’s what my younger son had to say:

(a) The 3d cube being cut by a (slightly thick) plane:

(b) The 4d cube being cut by the hyperplane:

This project was really fun, and, as I mentioned above, would also be a great programming project for kids learning linear algebra and trig. I’m 3d printing some of the shapes how, so playing with those shapes will be our project tomorrow!

## Kelsey Houston-Edwards’s hypercube video is incredible

The latest video that Kelsey Houston-Edwards released is one of the best math videos that I’ve ever seen:

This morning I used the video for a project with the boys. We watched (roughly) the first 11 minutes. I stopped the video before the Houston-Edwards revealed shapes associated with the 4D cube.

Here’s what the boys thought of the video:

Next we talked about how the rule for Pascal’s triangle related to the shapes in the video. This procedure is really fun to talk through with kids.

At the end of this part of the project we talked about the possible shapes that we would encounter in the 4th dimensional version of the problem.

Now that we had some of the basics sorted out for the 4th dimensional case, we tried to figure out the shapes. I won’t give away the answers in the text, but we were able to get the first one, but the 2nd one was pretty hard to see – the closest we got was that “it is sort of a twisting shape” 🙂

To wrap up we watched the remainder of Houston-Edwards’s video. The boys were surprised to learn what that final shape was.

At the end I reminded them of a prior project that was sort of similar – studying the so-called Prince Rupert Cube. Unfortunately the cube broke while I was reminding them about that project. Oh well . . . .

At this point I’m expecting Kelsey Houston-Edwards’s videos to be incredible, but she exceeds my expectations every time! Her videos are so much fun to share with kids – I can’t wait for the next one!

## A project inspired by Steve Phelps’s Dissection tweet

I saw a neat tweet from Steve Phelps today:

We’ve done a couple of projects on the Rhoombic Dodecahedron before – here are three of them:

Using Matt Parker’s Platonic Solid video with kids

A 3D Geometry proof with few words courtesy of Fawn Nguyen

Penrose Tiles and some simple 3D Variations

After seeing Phelps’s tweet I thought it would be fun to see if the boys remembered how to find the volume of the shape. So, I built one out of our Zometool set and asked them what they knew about the shape.

Here’s what my older son had to say:

Here’s what my younger son had to say:

I’m glad I saw Phelps’s tweet – it was fun to revisit some of these old projects occasionally. Also, it was a nice reminder of how easy it is to share 3d shapes with kids using a Zometool set.