It was the week of the state tests up here, so my kids had already spent quite a bit of time taking multiple choice tests this week. But one more test couldn’t hurt, right!

Here’s the introduction to the test – I love my younger son’s reaction and his enthusiasm!

After the introduction the boys sat down to take the test. Here’s what my younger son thought. It took him approximately 10 min to complete it – it is really fun to hear how he worked through it.

Here’s what my older son thought. He finished very quikcly – maybe in around 2 min. Luckily there was also a 10 question test linked in Patrick’s blog post, so I had him take that test, too:

Definitely a fun start to the weekend and a neat way for kids to get a peek at self referential ideas. Once you open that door, you can find some really amazing (and really strange!) mathematical ideas.

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!

[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.

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.

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:

Tonight I sat down with the boys to make sure they understood the problem. They noticed that half the numbers would have no powers of two – good start! After that observation they started down the path to solving the problem really quickly. In fact, my younger son thought that we might have a geometric series.

Since we covered a few ideas pretty quickly in the last video, so I stared this part of the project by asking them to give me a more detailed explanation for how they got the 1/2, 1/4, 1/8, . . . pattern in the last video. It turned out to be a little harder for them to give precise arguments, but they did manage to hit the main points which was nice.

At the end of this video my older son was able to write down the series that we needed to add up to solve the problem.

Now that we had the series, we had to figure out how to add it up. My guess was that they’d never seen a series like this, but my older son had a really cool idea almost immediately – rewrite the series!

The boys were able to sum the series in this new form – so yay!

At the end of of the last video my younger son said that he was surprised that the “expected value” wasn’t zero since zero was the most likely value. In this part of the video we talked a bit more about what “expected value” meant.

Once we had that I asked what I meant to be a quick question -> is the expected number of 3’s higher or lower. It turned out to be a longer conversation than I expected, though, because my older son was actually able to write down the answer!

Definitely a fun problem. I think it is fun for kids to see how to add up a series like the one in this project. I also think it is fun for kids to explore some of the basic ideas about primes that pop up in this problem.

As an aside, one other place where I’ve seen the series that came up here is in this post from Patrick Honner:

Today we explored the 2-dimensional idea in Kelsey Houston-Edwards’s video in 3-dimensions using 3d printing.

First, though, I asked the boys what they thought about the latest PBS Infinite series video:

Next we took a look at 6 different “spheres” in 3-dimensions which were defined using the different way of measuring distance Kelsey Houston-Edwards introduced in her video. It is really fun to hear kids talking about these shapes, and even more fun to be able to actually hold these shapes in your hand!

The last thing we did was look at the 6 spheres all together. My younger son noticed how much bigger the shapes got as you moved from the L-0.75 norm to the L-4 norm. My older son noticed that some of the shadows of the shapes looked pretty similar even though the shapes didn’t look the same at all:

This was a really fun project to prepare. It is really fun to show kids ideas from advanced math that the wouldn’t likely see in school. It is also really fun to hold these strange shapes in your hand! Lately we’ve been using the F3 program to help us make objects to print – it was a really lucky coincidence to see the new PBS Infinite Series video *after* I learned to use the new program. It took less than 5 minutes to make the .stl files for the 6 shapes 🙂

However, since we do not currently have a model of the 120-cell anywhere around the house, before starting the project tonight we took a look at two of Henry Segerman’s movies:

(1) Half of a 120-cell

and

(2) Toroidal Half 120-cell

With Segerman’s videos as background, we now calculated what “” would be for the 120-cell:

We approached the 600-cell the same way – starting with a video from Henry Segerman:

After that quick introduction we discussed the shape and calculated the value of “” for it. Turned out that it was the most 4-d spherical of the shapes that we looked at. That was a fun fact, and thinking about that fact caused my son to ask what a 4-d sphere actually looked like!

Well, I couldn’t end this week-long project with the question about the 4-dimensional sphere hanging in the air. We talked about the shape for about 5 min and then took the dog for a walk 🙂

As we walked up the street at the end of the walk my son turned to me and said:

“Wait a minute, all of the spheres would have to differ by infinitesimal amounts . . . . but, oh, there are infinitely many of them so I guess that’s ok.”

The project couldn’t have ended on a better note! Thanks to Patrick Honner for the great Pi day exercise which inspired this project. Thanks also to Henry Segerman for his videos about the 120- and 600- cells. I hope to own a few more of his 3d prints soon!

Unfortunately my older son didn’t remember the previous exercise – ha! Oh well, luckily we started with a quick review of the hyperdiamond and the rhombic dodecahedron:

After that quick talk about the 24-cell we returned to our whiteboard to talk about the value of “$\latex \pi$” for this shape. The “surface volume” and the “hyper volume” for the 24-cell turn out to be fairly simple numbers, and that but of luck gives us an easy value of “ for this shape!

So, one last project tomorrow. We’ll look at the 120-cell and the 600-cell. Can’t wait!

The main idea today is to calculate “” for the first three 4-dimensional platonic solids – the 5-cell, the 8-cell (aka the hypercube), and the 16-cell. A fun twist is that the 5-cell and the 16-cell have some 3d projections that are quite similar, but give quite different values for “”

So, we started with a super quick review of the 4d formula for and then took a look at the 5-cell. Although we didn’t go through the calculation, I liked my son’s guess that the hyper-volume of a 4-dimensional pyramid would be given by (1/4) * volume of base * height.

Next we looked at the 8-cell, or hypercube. Luckily this shape has really easy “surface volumes” and “hyper-volumes.” That allowed us to calculate “ exactly without too much difficulty – plus we got a little bit of exponent review 🙂

The last shape we looked at today was the 16-cell. This is the most difficult shape to understand, and understanding it is made even more confusing because we have a couple different 3-dimensional projects and they don’t look anything like each other! Also, as noted above, one of them looks a lot like the 5-cell.

It was fun to think about the “spherical-ness” of this shape prior to doing the calculation.

We are really having a lot of fun with this project. Tomorrow we’ll probably focus on the hyperdiamond because it is such a cool shape. Then we’ll talk about the 120-cell and the 600-cell for the grand finale 🙂