# Revisiting Grant Sanderson’s “fair division” video thanks to an Amy Hogan tweet

I saw a tweet from Amy Hogan this morning that reminded me of an old project:

The old project is here:

And Grant’s video is here:

Before we started the project today, I had my son watch Grant’s video. Here are some of his takeaways:

Now we tried to see if we could find a fair division of a string with two colors using only two cuts:

Finally, to wrap up we tried to see if we could do a fair division of a string with three colors using three cuts:

This is a fun and relatively easy project for kids, and also a way for them to see some really fascinating advanced math!

# Going back to James Tantons’ amazing Möbius Strip cutting project

James Tanton’s Solve This book is full of incredible math projects to do with kids:

Today we went back to do a project that we’ve looked at a few times before – cutting various versions of a Möbius Strip.

We started with a cylinder just to get going with an easy shape:

Next we moved on to a Möbius strip – but when we made this one we didn’t notice that we’d made a full twist rather than a half twist. So we sort of got two surprises. I’ve included this mistake to show that you do have to be a little careful when you do this project – it is easy to accidentally make the wrong shape:

Here’s the correction and the actual Möbius Strip:

Next was a shape where we had a 2nd little error – this time the tape held the final shape together a bit too much. In our video you can sort of guess what the actual shape will be, but I definitely encourage everyone to try this one out for yourself!

The 4th shape is where things really get interesting. For this and the next two shapes we are making a circular cut through a cylinder with a hole cut out of it. Each of the shapes has a interest set of twists in it – the shape in this video has a single half twist on one of the “arms”:

The 5th shape has half twists (going the same direction) in each of the arms:

The 6th shape has half twists going in different directions in each of the arms:

This really is an amazing project for kids. As our version of the project here shows (accidentally!), you do have to be careful with the preparation, but even with the couple of errors we had, it was still an incredibly fun morning.

# Having the boys try out some fun and surprising ideas about 360 and 720 degree rotations from Eric Weinstein’s interview with Roger Penrose

Yesterday I heard a terrific interview with Roger Penrose on Eric Weinstein’s podcast:

The podcast episode is here:

‘Today the boys and I talked through three fun to see, but maybe tricky to understand, ways that something need to rotate 720 degrees to get back to where it started.

We started by looking at a circle rotating around a second circle of the same size:

Next we looked at the famous “wine glass” problem. I originally wanted to color the water in the glass with food coloring, but chickened out!

Before going on to the Dirac Belt Trick, I showed the boys this really nice video showing the trick in a pretty unusual – and super fun – way:

After the video demonstration, I had the boys try the trick with a belt. At the end my old son made a connection between the belt trick and the complex numbers which was a nice and totally out of the blue surprise to me:

Anyone interested in physics should listen to Weinstein’s interview of Penrose – it is amazing. I was really happy to be able to pull out a few ideas from the interview to share with the kids today!

# Revisiting James Tanton’s paper cutting exercise

This week I’m going to be giving a talk at the math camp at Williams college. The talk this year is going to be based on an amazing paper cutting project that I learned from James Tanton’s book Solve This:

As that tweet from 2016 suggests, we’ve looked at these paper cutting ideas before:

Today I had both kids try out the project with two shapes. One purpose of today’s project was to remind me of the rough paper size we need to do this project (folding an 8 1/2 x 11 inch sheet of paper into thirds – so roughly 3×11 inch strip – worked pretty well). But I was also interested to see what the kids thought of the shapes because the results are so surprising!

Also, the snoring in the background is our dog – lol 🙂

(1) Older son shape 1:

(2) Older son shape 2:

(3) Older son shape 3:

(4) Older son shape 4:

Btw – Solve This is an amazing book. I see several used copies on Amazon right now, and I can’t recommend it enough!

# Sharing Cédric Villani’s amazing talk about John Nash’s work with my younger son

Yesterday I saw an absolutely incredible talk by Cédric Villani on youtube:

Although the talk is a public lecture and fairly accessible to anyone interested in math, it really isn’t aimed at kids. That said, Villani gives a beautiful description of the flat torus starting around 28:00 that I thought my younger son would find interesting. So, I had him watch that part of the video, then play a few rounds of Pac Man, and then we talked about the ideas. As always, it is really fun to hear a kid thinking through and describing ideas from advanced math:

# A fun experiment sharing Grant Sanderson’s Topology video with a kid

I saw a really neat new video from Grant Sanderson this morning:

We’ve actually looked at the ideas Grant is sharing here before, but my son didn’t remember:

Grant Sanderson’s “Fair Division” video shows a great math project for kids

For today I asked my younger son (in 7th grade) to watch the video and take some notes. After he finished we started taking about what he saw. He was interested in the Borsuk–Ulam theorem and also he thought the “stolen necklace problem” was pretty neat:

Next we talked about the proof of the Borsuk-Ulam theorem. I was really happy that most of the main ideas that Grant shared in his video stuck in my son’s mind.

We wrapped up by talking about the “stolen necklace” problem. We did a few examples about that problem and then had a fun discussion about the equation for a sphere. My son was curious about the difference between the boundary of the sphere and the all of the points inside the sphere. In particular, he was wondering why the equation for a sphere Grant used was $x^2 + y^2 + z^2 = 1$ and not $x^2 + y^2 + z^2 \leq 1$

From there we had an interesting discussion about dimension. I didn’t expect the conversation to go in that direction, but I guess you never know what a kid is going to take away from a video about some pretty advanced math ideas 🙂

# Using the Infinite Galaxy puzzle from Nervous System to talk topology with kids

The boys and I spent yesterday working on the new Infinite Galaxy Puzzle from Nervous System:

Having finished it, I thought a project talking about some of the math behind the puzzle would be really fun for the boys.

Since the front cover of the puzzle says that it was inspired by the Möbius strip, I started today’s project talking about that shape:

Next we talked about the puzzle and what geometric / topological properties it has. The interesting mathematical question here is whether or not the puzzle is a Möbius strip?

It turns out the puzzle is projective plane!!

We spent the last part of the project today talking about the projective plane and a few other similar shapes.

Even without any of the math, this new puzzle from Nervous System is a really fun challenge. The mathematical ideas behind the puzzle move it from the “fun puzzle” real to the “blow your mind” realm, though!

I’m so happy to have found one of these puzzles at the Nervous System open house last weekend. What an amazing way to share some introductory ideas from topology with kids!

# More math with bubbles

Bubbles were just in the air this week!

and last night flipping through Henry Segerman’s math and 3d printing book I found these bubble project ideas:

So I printed two of Segerman’s shapes overnight and tried out a new bubble project this morning.

I started with some simple shapes from our old bubble projects – what happens when you dip a cube frame in bubbles?

The next shape we tried was a tetrahedron frame:

Now we moved on to two of Segerman’s shapes. These shapes are new to the boys and they have not previously seen what bubbles will form when the shapes are dipped in bubble solution.

If you enjoy listening to kids talk about math ideas, their guesses and descriptions of the shape are really fun:

The second shape from Segerman we tried was the two connected circles. We actually got (I think) a different shape than I’d seen in Segerman’s video above which was fun, and the boys were pretty surprised by how many different bubble shapes this wire frame produced:

Definitely a fun project. I tried a bubble project for “Family Math night” with 2nd graders at my younger son’s elementary school last year. Kids definitely love seeing the shapes (and popping the bubbles).

# James Tanton’s incredible Möbius strip cutting project

Today we revisited one of my all time favorite math projects for kids (also **revisiting**) :

We did this project once before, but I don’t think the kids remembered it:

An absolutely mind blowing project from James Tanton

The project is relatively simple to set up – you have strips of paper and make 5 Möbius strip-like shapes.  If the short descriptions below aren’t clear, don’t worry, the videos have the “picture is worth 1,000 words descriptions

(1) An actual Möbius strip
(2) Same set up as making one Möbius strip, but you start with two strips of paper stacked on top of each other,
(3) A cylinder with a long oval cut out and a half twist on one of the strips left over after removing the oval.
(4) A cylinder with a long oval cut out and a half twist (in the same direction) in both of the strips left over after removing the oval.
(5) Same as (4) but the twists are in opposite directions.

Then you cut the shapes. In (1) and (2) you cut completely along the center line. In (3), (4), and (5) you cut around the oval.

What shape are you left with after the cutting?

(1) Cutting a Möbius strip

(2) Cutting two strips of paper folded into a Möbius strip

(3) Cutting a cylinder with an oval removed with one half twist

(4) Cutting a cylinder with an oval removed with two half twists in the same direction

(5) Cutting a cylinder with an oval removed with two half twists in opposite directions

# Sharing Sam Hansen’s “Knotty Helix” podcast with kids

The latest Relatively Prime podcast is fantastic:

The short description from the podcast’s website is:

“Sure DNA is important, some might even claim it is absolutely integral to life itself, but does it contain any interesting math? Samuel is joined by UC-Davis Professor of Mathematics, Microbiology, and Molecular Genetics Mariel Vazquez for a discussion proves conclusively that mathematically DNA is fascinating. They talk about the topology of DNA, how knot theory can help us understand the problems which occur during DNA replication, and how some antibiotics are really pills of weaponized mathematics.”

Since it is only 20 min long, I thought it would be fun to share with the boys. We listened to it in the car when we went out to breakfast this morning. Upon returning home I asked the kids what they thought and what were somethings they learned. Here’s what they had to say:

Next we looked at an interesting process described in the podcast. That process was an example by Mariel Vazquez of how you can go from a link with 6 crossings to two unlinked circles.

The process in the podcast seems simple – maybe even obvious – but I think that the process is actually much more subtle than it seems listening to it.

Here we followed the steps to go from the trefoil knot to the two unlinked circles. I think the ideas we followed here are a great way for kids to explore the process described in the podcast:

The next thing we looked at was the idea that when you cut a loop with an even number of twists in half, the halves would be linked. We took a long strip of paper and gave it 6 twists, taped the ends together, and then cut it down the middle.

I fast forwarded through the taping and cutting part, but forgot to remove the sound. Sorry for the “Alvin and the Chipmunks” middle part of this video.

I think both the podcast and the follow up projects are a great way for kids to explore some math ideas that wouldn’t normally be part of a school curriculum.

We’ve done a few other projects with knots and with paper cutting. Here’s a link to those collections:

Our knot projects

A collection of some of our paper cutting (and folding) projects

It was really neat to hear about how knot theory applies to biology.