Fun with Möbius Strips

The interesting math topic that I saw in a couple of different places over the last two days involves Möbius strips, and a question from my youngest son about the Möbius strip this morning led to a fun conversation.

The thing that got him thinking about Möbius strips was this 3D printing page about the Umbilic Torus:

We followed the directions on this page yesterday evening and printed a small triangular cross section Möbius strip.  It is a really neat shape and my younger son had a lot of fun playing with it yesterday before he went to bed.  I hope we can figure out how to print an actual Umbilic torus, but we don’t yet have that level of 3d printing skill just yet.

After he went to bed I saw this wonderful video posted by Steven Strogatz on twitter:

I was really happy to see Strogatz’s post for a couple of different reasons.  First, several months ago someone else had posted about the double Möbius strip in association with a Martin Gardner celebration and I’d not been able to find the link again.  Unfortunately I’d misunderstood the setup so it was nice to see where I’d gone wrong.  Second, having just spent half an hour talking about 3D Möbius strips the exercise in this video was a great natural extension (which I’ll probably do with the boys this weekend).

With that background, I was really happy to have my son ask this question today when he got up:

“What happens if you do two half twists instead of one when you make a Möbius strip?”

That led to this fun little discussion which shows a surprising result from geometry:

This is one of my favorite examples in math!

Of course, no discussion of Möbius strips is complete without showing Vi Hart’s amazing “Wind and Mr. Ug” video:

Such a fun topic for kids.