Playing with Pascal’s triangle and angles hidden in cubes

In April 2018 I saw a great Numberphile video with Federico Ardila:

The project that we did after seeing that video is here:

Federico Ardila’s Combinatorics aand Higher Dimensions video is incredible

This week my younger son was learning about coordinates in 3 dimensions in his precalculus book and I though it would be fun to revisit some of the ideas about cubes from Ardila’s video.

We started by looking at cubes in 0 to 4 dimensions and discussing how we could see Pascal’s triangle hiding in the cubes:

In the last video we got a little hung up on the 4-dimensonal cube, so for the next part of the project we looked at the coordinates of the vertices of the various cubes to see if that could help us see Pascal’s triangle in the 4d cube.

Next we moved on to looking at the angles made by the long diagonal in the various cubes. This exercise was particularly nice since my younger son has been learning a little trig and my older son has been learning a bit of linear algebra.

For the final part of the project we looked at the 4-d cube. Here are zometool shape isn’t really helping us see the long diagonal. My younger son did a really great job seeing the pattern in the right triangles with the long diagonal. He also noticed the amazing fact that there is a 30-60-90 triangle hiding in a hypercube!

My older son was also able to find the same angle using ideas from linear algebra:

Definitely a fun project. It is fun to introduce coordinates not just for 3 dimensions, but for all dimensions at the same time. There’s also an enormous amount of fun math hiding in the seemingly simple idea of n-dimensional cubes, which makes this project sort of doubly fun!

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Visualizing the 5d permutohedron with kids

Last night as part of a linear algebra project I was doing with my older son, we found out that you can orient a 4d permutohedron in 3 space so that all of the vertices have integer coordinates:

Today I wanted to explore that idea a bit more and also include my younger son. So, I thought it would be fun to see if we could find a way to see what the 5d permutohedron looks like by looking at slices of it in 4d.

I started by reviewing the 3d permutohedron and how it is embedded in 2 dimensions. It was nice to go back to the beginning here – especially so that we could explore how slicing with lower dimensional slices works.

Next we tried the same “visualization by slicing” idea with our 4d permutohedron embedded in 3 dimensions:

Finally, and sorry this one is long, we got to the heart of today’s project. Here we’ll be using some code I wrote in Mathematica to view 3d slices of the 5d permutohedron emedded in 4d space. It is close to a miracle that I was able to get these visualizations to work correctly – maybe the extra hour this morning helped! It was super fun to hear the boys talk about what they saw with these shapes:

Talking about angles in Platonic solids

My younger son wanted to do a Zometool project today and since my older son is currently learning about the dot product, I thought it would be fun to talk about angles in some platonic solids.

This idea turned out to be one that was better in my mind than it was in practice – ha! – but it was still a nice project even though it got a bit messy.

We started by talking about angles in a cube:

Next we moved to the octahedron:

Here we go through the steps to calculate the angle between two faces in the octahedron:

Finally, we wrap up by looking at the fun surprise that a hypercube has a 30-60-90 triangle hiding in it! My younger son got a little confused about how to find the lengths of some of the vectors we were looking at, so we went slow. It is really fun to see how some relatively simple ideas let you explore hard to visualize objects like a 4-dimensional cube!

Exploring an amazing tweet from John Carlos Baez with my younger son

I saw an incredible tweet from John Carlos Baez last week:

Here’s the picture in case the tweet isn’t embedding all that well for you:

Baez

I thought exploring some of these shapes would make a great project for kids, so I began by asking my son (in 7th grade) for his thoughts on the shapes he was seeing:

Next we built a few of the cubes from our Zometool set and talked about some of the shapes. First, though, I asked my son to give his definition of what an n-dimensional cube was:

Finally, we played around by a different version of a 4-dimensional cube – “Hypercube B” by Bathsheba Grossman. This amazing version of the hypercube makes amazing shadows and my son was able to find a projection that was a little closer to the projection of the 4d cube in Baez’s tweet:

Also, here’s a video I made a while back showing some other (almost freaky) 2d projections from our Zometool model of Hypercube B:

Definitely a fun project – thanks to John Carlos Baez for sharing some of his ideas about higher dimensional cubes on twitter!

My talk at the 2018 Williams College math camp

[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:

Counting geometric properties in 4 and 6 dimensionsf

(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:

Kelsey Houston-Edwards’s hypercube video is incredible

One more look at the hypercube

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:

Here are the 2d slices of the 3d cube:

Here are the 3d slices of the 4d cube:

(4) Finally, we looked at the connection between cubes and combinatorics

I learned about this connection from this amazing video from Numberphile and Federico Ardila:

Here is the project I did with my older son after seeing Ardila’s video:

Federico Ardila’s Combinatorics and Higher Dimensions video is incredible!

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:

Jim Propp’s essay Time and Tesseracts

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.

Counting in 4 dimensions

Yesterday we did a neat project based on problem #12 from the 2015 AMC 8:

Problem12

That project is here:

A great counting problem for kids from the 2015 AMC 8

Then I got a nice comment on the project from Alison Hansel:

So, for today’s project we extended the problem from yesterday to 4 dimensions.

Here’s the introduction and a quick reminder of yesterday’s problem. I had both boys review their solutions and then we began to discuss how to approach the same problem in 4 dimensions.

Next we dove a bit deeper into how to approach the 4 dimensional problem. They boys thought a bit about the symmetry that a 4d cube would have and at the end (after a long and quiet pause) my younger son thought that looking at how a square turns into a cube might help us.

In studying how a square transforms into a cube, the critical idea is how 4 edges turn into 12 edges. This video is a little on the long side, but I think the discussion is really interesting. By the end the boys have found the main idea for how to count edges as you move up in dimension.

Next I brought out Henry Segerman’s 4-d cube model and compared the model to the ideas we’d developed up to this point.

An important idea from earlier in this project was that my older son thought that each edge of the 4d-cube would be part of two 3d cube “faces”. Using the model we were able to see that, in fact, each edge is part of 3 cubes.

Finally – with the 4 pieces of prep work behind us! – we were able to answer the AMC 8 question about a 3d cube in 4 dimensions. So . . . how many pairs of parallel edges does a 4d cube have? The answer is 112 🙂

Thanks to Alison Hansel for the great suggestion for how to extend yesterday’s project. I think her idea makes a great way to introduce kids to some simple ideas in 4d geometry.

Extending our Alexander Bogomolny / Nassim Taleb project from 3 to 4 dimensions

Last week I saw really neat tweet from Alexander Bogomolny:

The discussion about that problem on Twitter led to a really fun project with the boys:

A project for kids inspired by Nassim Taleb and Alexander Bogomolny

That project reminded the boys about a project we did at the beginning of the summer that was inspired by this Kelsey Houston-Edwards video:

Here’s that project:

One more look at the Hypercube

For today’s project I wanted to have the boys focus on the approach that Nassim Taleb used to study the problem posed by Alexander Bogonolny. That approach was to chop the shape into slices to get some insight into the overall shape. Here’s Taleb’s tweet:

Taleb1

So, for today’s project we followed Taleb’s approach to study a 4d space similar to the space in the Bogomolny tweet above. The space is the region in 4d space bounded by:

|x| + |y| + |z| \leq 1,

|x| + |y| + |w| \leq 1,

|x| + |w| + |z| \leq 1, and

|w| + |y| + |z| \leq 1,

To start the project we reviewed the shapes from the project inspired by Kelsey Houston-Edwards’s hypercube video. After that we talked about the equations we’d looked at in the project inspired by Alexander Bogomolny’s tweet and the shape we encountered there:

Next we talked a bit about the equations that we’d be studying today and I asked the boys to take a guess at some of the shapes we’d be seeing. We also talked a little bit about absolute value which briefly caused a tiny bit of confusion.

The next part of the project used the computer. First we reviewed Nassim Taleb’s approach to studying the problem posed by Alexander Bogomolny. I think it is really useful for kids to see examples of how people use mathematical ideas to solve problems.

The 2d slicing was a fascinating way to approach the original 3d problem. We’ll use the same idea (though in 3d) to gain some insight on the 4d shape.

One fun thing about this part of the project is that we encountered a few shapes that we’ve never seen before!

Finally, I revealed 3d printed copies of the shapes for the boys to explore. They immediately noticed some similarities with the hypercube project. It was also really interesting to hear them talk about the differences.

At the end, the boys think that the 4d shape we encountered in this project will be the 4d version of the rhombic dodecahedron. We’ve studied that shape before in this project inspired by a Matt Parker video:

Using Matt Parker’s Platonic Solid video with kids

I don’t know if we are looking at a 4d rhombic dodecahedron or not, but I’m glad that the kids think we are 🙂

It amazes me how much much fun math is shared on line these days. I’m happy to have the opportunity to share all of these ideas with my kids!