Klein Bottles and Möbius strips

My younger son has been interested in Klein bottles for the last couple of days. Not sure why, but we’ve printed out three Klein bottle models from Thingiverse that he’s enjoyed playing around with:

The links to each of the three models on Thingiverse are below:

Voronoi Klein Bottle by MadOverlord

Steampunk Klein Bottle by adamwebb

Klein bottle with base by mpatoulachik

My son’s sudden interest in these models made me think of a fun topology project we could do this morning – showing the boys how to represent the Klein bottle + some other shapes as a single sheet of paper. The basics of the idea is on this Wikipedia page:

The Fundamental Polygon

I started with a reminder of how to make Möbius strips. This simple idea got the boys doing something that they already knew how to do, but gave me the opportunity to introduce the idea of identifying arrows on a sheet of paper:

Next we moved on to how to make a torus. Since this is the only shape that is really easy to see, I had the boys bring over some silly putty to help make the shape. Only as I’m writing this am I realizing that the silly putty would have been pretty helpful for the sphere, too – oh well! Even with the silly putty, it took a couple of tries to make the shape and for the boys to really see what the shape looks like. Eventually, though . . . “a doughnut!”

The next shape we talked about was the sphere. This was a very tough one for them to see and understand. I think it was hard for them to see that there was an inside and an outside because they sort of got stuck thinking about triangles. By luck the egg-shaped silly putty case was nearby and that helped them see the geometry a little. My guess is that understanding why the square diagram for the sphere does represent a sphere is the key to understanding these diagrams. It is sort of a mind bending exercise when you see it for the first time!

The next shape was the Klein bottle. Now things get pretty strange, but the power and simplicity of these square diagrams also starts to become clear. I personally find it very hard to visualize how the square folds up into a Klein bottle, though the models that we’ve printed that let you see inside help with the visualization. I also used a Möbius strip to help them see a second way to visualize the construction of the Klein bottle. Not sure the kids got all the way to understanding how to make a Klein bottle from a square, but I think they took the first step!

The last shape we studied was the projective plane. I have never been able to visualize this shape, so for me the square with edges identified is the easiest way to understand it. In this last video we talk about this square and talk about why the shape it describes is particularly hard to visualize:

So, a fun morning talking about some introductory topology. Not sure what the follow up project to this one would be, but this is still a fun one just for trying to help them wrap their minds around all of these shapes. Being able to hold the Klein bottle in your hand makes these shapes a little easier to understand, so this project is a nice little demonstration of the educational value of a 3D printer, too.