Why cutting a Möbius strip is so weird

An explanation for why cutting a Möbius strip down the middle keeps it connected.

Resources to learn more and other interesting notes:

The boundary of a Möbius strip is a circle. If you make 3 half-twists, you still get a Möbius strip, but the boundary is knotted into what's called a "trefoil knot." We call a surface that bounds a knot a Seifert surface.

More about Seifert surfaces:

Corrections:

[zeleniyslon]

Thanks for the video. I tried thinking about the cut a third of the way in terms of the diagram and at first, couldn't understand why it would be different from the cut down the middle, but then realized that I need to draw it symmetrically to the center line since the cut ends don't meet on one trip through the strip. And it's actually evident from one strip being twice as long as the other one. Though I wonder if one can see that the strips must stay "together" from the diagram, I don't see a way from the get-go.

As for the strip with a full twist, if we consider just the edges it is as if we are wrapping one rope around the other and then connecting one end of each rope to its other end. We thus get two "loops" that go through each other, and since we don't cut them, they'll stay together.

I have one more idea for the video. Just today I encountered a mention of platonic solids and wanted to see if I could come up with a proof that there are really just 5 of them. Would be cool to see some ways to prove this. Also, I wonder how many possible solids are there in higher dimensions

[chengong]

Yea I get the explanations, but it’s still, black magic. I mean when you cut it down the middle you can’t possibly be exactly on the mid line, you have to be a little off the middle, but you can never accidentally turn it into two rings can you? How’s that possible? Where precisely is the exact place to cut for it to turn into two rings? And what if I cross that exact place while cutting? What happens then?

[fibbooo]

To answer the end of the video: lets imagine the 2 half twist diagram as two 2 d squares, with the edges linked in opposite orientation in the middle and the outside.

When we cut in the middle, the top edge on one square and the bottom edge of the other square make one edge of the resulting shape, bound by the cut in the middle. So we’ll have 2 shapes. They are interlocked because the “flip” in the middle prevents them from moving past each other.

I don’t think I did a good job explaining my reasoning! I gotta practice my topological writing

[unuboldlakechoros]

Not me watching this to understand my manifolds lectures

[SPVLaboratories]

cool illustrations. best of luck on the algorithm

[blucat]

Thank you for this great explanation. I wish I had seen it when I was a kid. I used to be smart, now I'm 57 and forget my own phone number!!
I have a question for you, similar, maybe you could do the video on it.
If the formula for the surface area of a circle is πr2, why isn't the formula for the volume of a sphere πr3? Doing the visuals would require some CGI though, still, it would be good. Cheers. 🙂