## Calculating the volume of our rhombic dodecahedron

Yesterday we did a fun project involving a rhombic dodecahedron:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

At the end of that project we were looking carefully at how you would find the volume of a rhombic dodecahedron in general. Today I wanted to move from the general case to the specific and see if we could calculate the volume of our shapes. This tasked proved to be much more difficult for the boys than I imagined it would be. Definitely a learning experience for me.

Here’s how we got going. Even at the end of the 5 min here the boys are struggling to see how to get started.

So, after the struggle in the first video, we tried to back up and ask a more general question -> how do we find the volume of a cube?

Now we grabbed a ruler and measured the side length of the cube. This task also had a few tricky parts -> do we include the zome balls, for example. But now we were making progress!

Finally we turned to finding the volume of one our our 3d printed rhombic dodecahedrons. We did some measuring and found how many of these shapes it would take to fill our zome shape and how many it would take to fill a 1 meter cube.

So, a harder project than I expected, but still fun. We’ve done so much abstract work over the years and that makes the concrete work a little more difficult (or unusual), I suppose. I’m happy for this struggle, though, since it showed me that we need to do a few more projects like this one.

## A project for kids inspired by Nassim Taleb and Alexander Bogomolny

I woke up yesterday morning to see this problem posted on twitter by Alexander Bogomolny:

About a two months ago we did a fun project inspired by a different problem Bogomolny posted:

Working through an Alexander Bogomolny probability problem with kids

It seemed as though this one could be just as fun. I started by introducing the problem and then proposing that we explore a simplified (2d) version. I was excited to hear that the boys had some interesting ideas about the complicated problem:

Next we went down to the living room to explore the easier problem. The 2d version, $|x| + |y| \leq 1$, is an interesting way to talk about both absolute value and lines with kids:

Next we returned to the computer to view two of Nassim Taleb’s ideas about the problem. I don’t know why the tweets aren’t embedding properly, so here are the screen shots of the two tweets we looked at in this video. They can be accessed via Alexander Bogomolny’s tweet above (which is embedding just fine . . . .)

The first tweet reminded the boys of a different (and super fun) project about hypercubes inspired by a Kelsey Houston-Edwards video that we did over the summer:

One more look at the Hypercube

The connection between these two projects is actually pretty interesting and maybe worth an entire project all by itself.

Next we returned to the living room and made a rhombic dodecahedron out of our zometool set. Having the zometool version helped the boys see the square in the middle of the shape that they were having trouble seeing on the screen. Seeing that square still proved to be tough for my younger son, but he did eventually see it.

After we identified the middle square I had to boys show that there is also a cube hiding inside of the shape and that this cube allows you to see surprisingly easily how to calculate the volume of a rhombic dodecahedron:

Finally, we wrapped up by using some 3d printed rhombic dodecahedrons to show that they tile 3d Euclidean space (sorry that this video is out of focus):

Definitely a fun project. I love showing the boys fun connections between algebra and geometry. It is also always tremendously satisfying to find really difficult problems that can be made accessible to kids. Thanks to Alexander Bogomolny and Nassim Taleb for the inspiration for this project.

## Steve Phelp’s 3d pentagon

Sorry that this post is written in a bit of a rush . . . .

I saw a neat tweet from Steve Phelps earlier in the week:

The shape sort of stuck in my mind and last night I finally got around to making two shapes inspired by Phelp’s shape. My shapes are not the same as his – one of my ideas for this project was to see if the boys could see that the shapes were not the same.

So, we started today’s project by looking at the two shapes I printed overnight. As always, it is really fun to hear kids talk about shapes that they’ve never encountered before.

Next we looked at Phelp’s tweet. The idea here was to see if the boys could see the difference between this shape and the shapes that I’d printed:

Finally, we went up to the computer so that the boys could see how I made the shapes. Other than some simple trig that the boys have not seen before, the math used to make these shapes is something that kids can understand. We define a pentagon region by 5 lines and then we vary the size of that region.

I’m not expecting the boys to understand every piece of the discussion here. Rather, my hope is that they are able to see that creating the shapes we played with today is not all that complicated and also really fun!

This was a really fun project – thanks to Steve Phelps for the tweet that inspired our work.

## 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.

## Sharing a Craig Kaplan post with kids part 2

Yesterday we used a recent post from Craig Kaplan as a way to talk a little bit about algebra and geometry:

Here’s that project:

Sharing a Craig Kaplan post with kids

After the project I printed 12 of the pentagons and had the kids play with them today. See Kaplan’s post for some historical notes about the pentagon. The historical importance is probably too advanced for kids to appreciate, but what they can appreciate is that this pentagon can be surrounded in multiple ways. I had the boys play around to see what they could find.

Here’s what my younger son found:

Here’s what my older son found:

This project was really neat. I think making shapes like the one in Kaplan’s post is a great way for kids to review (or even get introduced to!) both equations of lines and some elementary geometry. Also, as always, it is extremely fun for kids to explore ideas that are interesting to professional mathematicians 🙂

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## Sharing a Craig Kaplan post with kids

I saw the latest post from Craig Kaplan via a tweet from Patrick Honner:

The picture in the middle part of the post looked like something that kids could understand:

For our project today I thought it would be fun to talk about how to make the polygon tile in the above picture. After we understand how to describe that polygon, we can 3d print a bunch of the tiles and talk more about the idea of “surrounding a polygon” with these tiles tomorrow.

This project is a fun introduction to 2d geometry (and especially coordinate geometry) for kids. We also use the slope / intercept form of a line when we make the shape.

We got started by looking at Kaplan’s post:

Next we began to talk about how to make the shape – the main idea here involves basic properties of 30-60-90 triangles. My older son was familiar with those ideas but they were new to my younger son.

We also talk a little bit about coordinate geometry. The boys spend a lot of time discussing which point they should select to be the origin.

In the last video we found the coordinates of 3 of the points. Now we began the search for the coordinates of the other two. We mainly use the ideas of 30-60-90 triangles to find the coordinates of the first point.

The 2nd point was a bit challenging, though:

The next part of the project was spent searching for the coordinates of the last point. The main idea here was from coordinate geometry -> finding the coordinates of the middle of the square. The coordinate geometry concepts here were difficult for my younger son but we eventually were able to write down the coordinates of the final point:

We were running a little long in the last video, so I broke the video into two pieces. The last step of the calculation is here:

After finding all of the coordinates we went upstairs to make the shape on Mathematica. We used the function “RegionPlot3D” that allows us to define a region bordered by a bunch of lines. Below is a recap of the process we went through to make the shape and a quick look at the shapes in the 3d printing software:

This isn’t our first 3d printing / tiling project. Some prior ones are linked in a project we did last month after seeing an incredible article by Evelyn Lamb:

Evelyn Lamb’s pentagons are everything

I’m excited with the boys to play with the tiles from Kaplan’s post tomorrow.

## 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!

## Today I got one step closer to a long-term goal

One of the math mountains that I’ve always wanted to try to climb is to find a way to explain to kids why 5th degree polynomials can’t be solved in general.

The “one step closer” came from a comment by Allen Knutson on one of our projects on John Baez’s “juggling roots” tweet. Here’s the tweet:

Here are the two recent projects that we’ve done after seeing that tweet. Knutson’s comment is at the end of the first post:

Sharing John Baez’s “juggling roots” tweet with kids

Sharing John Baez’s “juggling roots” post with kids part 2

The comment pointed me to a video that shows how the “juggling roots” approached can be used to show that there is no general formula for finding the roots of a 5th degree equation:

The neat thing about the combination of this video and Baez’s post is that you can see some of the ideas from the video in the “juggling roots” gifs in the post.

Tonight I used some of the 3d prints of the juggling roots that I’ve made in the last few days to talk about the ideas a bit more and then we watched just a few minutes of the video.

We started with with a print that I accidentally made twice – but luckily the two prints give us a way to view the juggling roots through two cycles:

Next we looked at a different print to see a different juggling roots pattern. Here I was trying to set up the idea that the roots can move around in different ways. The way those different movements interact is the key idea in the video that Allen Knutson shared.

Finally, we went upstairs to watch a little bit of the video. Sorry for the sound issues, I don’t know why I left the sound on in the video. I mainly wanted the boys to see a different view of the juggling roots and I told them that the video gave the explanation for why 5th degree polynomials can’t be solved in general:

So, although I don’t quite have a full explanation of 5th degree polynomials for kids – I feel like I took a giant step towards getting to that explanation today. It is an extra happy surprise that 3d printing is going to come into play for that explanation!