Henry Segerman’s Flat Torus

[Sorry for writing this one up at one million miles per hour. We are seeing Star Wars at 9:30 am today and I wanted to get this one published before we left. Hopefully the fun of this amazing shape outweighs the quick and unedited writing.]

Saw another amazing post from Evelyn Lamb a few weeks ago:

Can we make a flat torus without corners? https://t.co/ovItZHVOgU pic.twitter.com/BqQBpAtm0R

— Evelyn Lamb (@evelynjlamb) December 8, 2015

In that post she showed this video by Henry Segerman:

 

I decided to buy this shape from Shapeways and do a project with the boys. Here’s where you can order it:

Henry Segerman’s Flat Torus on Shapeways

Before starting the actual project this morning I had the kids explore the shape on their own. Although these individual explorations were a little more difficult than I was expecting I’m glad that we did them because it gave the kids a bit more familiarity with the shape. The videos from that part of the project are here:

Background for our project on Henry Segerman’s Flat Torus

So, with that background out of the way, we began today’s project by talking about how to make Möbius strips. The discussion to a little detour when I asked them to describe how to make the shape, but it was a cool and important detour. It is neat to hear kids talk about complicated shapes!

 

Next we talked about how to make a torus from a single sheet of paper. The kids had a hard time imagining exactly how a square would fold up into a torus, but that’s sort of the point of the project! At the end we looked at how the shape was sort of like a bagel, but sort of not like a bagel!

 

For the 3rd part of our project we looked more carefully at Henry’s shape an how a flat torus was different than the bagel.

 

The last part of the project was a quick discussion of the Klein Bottle and the projective plane, which we can also understand from a single square piece of paper. The boys were really interested in the ideas here, and especially how difficult it was to make the Klein bottle in three dimensions! They did make a connection between the Klein bottle and the Möbius strip, which was really cool to hear.

 

So, a super fun project exploring some really complicated shapes. Infinite thanks to Evelyn Lamb and Henry Segerman for inspiring this project!

Background for our project on Henry Segerman’s flat torus

This videos here are just background for a project on Henry Segerman’s flat torus. I’ll link that project after I write it up.

Before diving into that project, though, I wanted the kids to experience the shape without knowing what it was. It is both an engineering and math challenge.

My older son had a tough time, but made the folded up shape eventually. That it took nearly 15 minutes to make sense of this shape gives a little peek at how difficult this shape is to understand:

 

 

 

My younger son was also puzzled about what to do with this shape. He even wondered if it was hard to understand because it was 4 dimensional 🙂 At the end he thought it was a torus, which was cool: