## 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.

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

Yesterday I saw this incredible tweet from John Baez:

We did one project with some of the shapes this morning:

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

The tweet links to a couple of blog posts which I’ll link to directly here for ease:

John Baez’s “Juggling Roots” Google+ post

Curiosa Mathematica’s ‘Animation by Two Cubes” post on Tumblr

The Original set of animations by twocubes on Tumblr

Reading a bit in the comment on Baez’s google+ post I saw a reference to the 3d shapes you could make by considering the frames in the various animations to be slices of a 3d shape. I thought it would be fun to show some of those shapes to the boys tonight and see if they could identify which animated gif generated the 3d shape.

This was an incredibly fun project – it is amazing to hear what kids have to say about these complicated (and beautiful) shapes. It is also very fun to hear them reason their way to figuring out which 3d shape corresponds to each gif.

Here are the conversations:

(1)

(2)

(3)

(4)

(5)

(6) As a lucky bonus, the 3d print finished up just as we finished the last video. I thought it would be fun for them to see and talk about that print even though (i) it broke a little bit while it was printing, and (ii) it was fresh out of the printer and still dripping plastic 🙂

The conversations that we’ve had around Baez’s post has been some of the most enjoyable conversations that I’ve had sharing really advanced math – math that is interesting to research mathematicians – with kids. o

## Comparing Sqrt(x^2 + y^2) and ( Sqrt(x^2) + Sqrt(y^2) )

Last week we used 3d printing to compare $(x + y)^2$ and $x^2 + y^2$:

That project is here:

Comparing x^2 + y^2 and (x + y)^2 with 3d printing

My younger son is still sick today and not able to participate in a math project, so I chose a slightly more algebraically complicated comparison to look at with just my older son -> $\sqrt{x^2 + y^2}$ and $\sqrt{x^2} + \sqrt{y^2}$

Here’s what the shapes look like:

I started the project by reviewing the original project in this series just to remind my son about how we thought about the 3d surfaces in the prior post. He remembered most of the ideas, fortunately, so the introduction was fairly quick.

After the introduction we talked about some basics of the algebra we were going to encounter in this project, namely that $\sqrt{x^2} = |x|$. This part all by itself is a difficult concept to understand and the bulk of the video below was spent talking about it.

/

With the difficult part of the algebra behind us we moved on to talking about the surface $z = x^2 + y^2$. What does this surface look like?

I really enjoyed the discussion here – the question is actually a pretty challenging one for a kid to think through.

/

Next we tried to figure out what the surface $z = \sqrt{x^2} + \sqrt{y^2}$ would look like.

I think it takes a while to get used to working with graphs of the square root function. My son struggled a bit here to figure out the shape here. Hopefully that struggle helped him

/

Now I revealed the shapes and let my son discuss the properties of the shapes now that he could hold them in his hand. There were a few surprises, which was nice 🙂

/

I’m really happy about this series of projects. It is fun to explore the variety of ways that 3d printing can help kids explore math.

## Learning 3d geometry with Paula Beardell Krieg’s pyramids

Earlier in the week we got a nice surprise when we received a fun little pyramid puzzle from Paula Beardell Krieg:

Our initial project using the shapes is here:

Playing with an amazing present from Paula Beardell Krieg

I thought a follow up project would be fun, so I decided to try out a basic exploration in 3d geometry. The goal was to make these shapes ourselves using Mathematica, then to 3d print them, and finally to play with the new shapes to see that they were indeed the same.

We started by talking about the shapes in general and see if we could identify some very specific properties of the shape using coordinate geometry:

Next we talked about how to describe the planes that formed the boundary of the shape. It was fun hearing my 5th grader try to figure out how to describe the planes (and regions) we were studying here. One other challenge here is that we were also trying to describe the 3d regions above and below these planes.

Now came the special challenge of finding a mathematical way to describe the hard to describe plane in the shape. I had to guide the discussion a bit more than I usually do here, but the topic of finding the equation for a plane is pretty advanced and something that kids have not seen before.

Having written down the equations, we went up to look at the Mathematica code I’d used to make the shapes. The boys were able to see that the first shape had exactly the same equations we’d written down, and they were able to see that the equations for the 2nd shape were not any more difficult.

The shapes printed overnight and we had an opportunity to play with them this morning. It is pretty neat to hear them compare the shapes and see that, indeed, the shapes we made are really the same as the shapes Paula sent us.

So, there’s quite a lot we can study with Paula’s shapes. You’ve got the potential to study folding patters, basic 3d geometry, the volume formula for a pyramid, and even 3d printing! Fun how such a seeming simple idea can lead you in so many different directions.

## A fun shape for kids to explore: the Permutohedron

I learned about permutohedrons from a comment by Allen Knutson on a prior blog post. See the first comment here:

A morning with the icosidodecahedron thanks to F3

I prnted the shape from Thingiverse and it was amazing!

“Permutahedron” by PFF000 on Thingiverse

We started the project today by examining the shape and comparing it to a few other shapes we printed. The comparison wasn’t planned – the other shapes just happened to still be on the table from prior projects . . . only at our house 🙂

Next we talked about permutations and the basic idea we were going to use to make the permutohedrons. We drew the 1 dimensional version on the whiteboard and talked about what we thought the 2 dimensional version would look like.

We used our zometool set to make a grid to make the 2 dimensional permutohedron. Lots of different mathematical ideas for kids in this part of the project -> coordinate geometry, permutations, and regular old 2d geometry!

Next we went back to talk about how PFF000’s shape was made. Here’s the description on Thingiverse in case I messed up the description in the video:

“The boundary and internal edges of a 3D permutahedron.

The 4! vertices are given by the permutations of [1, 3, 4.2, 7], with an edge connecting two vertices if they agree in two of the four coordinates. The 4D vertices live in a 3D hyperplane, namely the sum of the coordinates is 15.2.

This part of the project was a little longer, but worth the time as both the simple counting ideas on the shape and the combinatorial ideas in the connection rules are important ideas:

Finally we wrapped up by taking a 2nd look at the shape and also comparing it to Bathsheba Grossman’s “Hypercube B” which was also still laying around on our project table!

This was a really fun project that brought in many ideas from different areas of math. I’m grateful to Allen Knutson for the tip on this one!

## A morning with the icosidodecahedron thanks to F3

A few weeks a go I saw this shape in a display case at the MIT math department:

The shape is mislabled, unfortunately, it is an icosidodecahedron. We’ve already done a few projects based on the shape. Last week’s project is here:

A zometool follow up to our Cuboctohedron Project

And there’s our fun Zometool Snowman, too, where the icosidodecahedron is the head:

The name suggests that it is made from a combination of an icosahedron and a dodecahedron – but how?