Twist in the tale of the Möbius strip

livemint.com:

NEW DELHI : Remember your first Möbius strip? I don’t. But I remember, every time I made one, a sense of wonder at what was about to happen. A two-sided strip of paper, just like every other, suddenly turns one-sided.

Two distinct edges, and suddenly they’re one. Then you should wield a pair of scissors on your Möbius strip. Cut down the middle and you have something unexpected and fascinating.

Cut about a third of the way in from the edge, and as you go around the strip, you’re suddenly a third of the way from the other edge, and when done, you have something else unexpected and fascinating. I won’t tell you what these two results are—best to marvel at on your own. But I defy you to predict what the cuts will do.

So, what are these marvellous strips? Take a long, narrow strip of paper—perhaps an inch wide, a foot long. Make a loop with it, bringing the ends together. But here’s the kicker: just before you join the ends, twist one of them so the top is bottom and the bottom is top.

Now join the ends. What’s in your hands is an authentic Möbius (often written Moebius) strip. As you admire your handiwork, consider that as simple as this device is, it was not “discovered" until the mid-19th century.

I wonder about that “discovery", though. Surely people had made paper loops before, even put twists in them, inadvertently or otherwise. Thus, it isn’t the strip itself that the German mathematicians August Ferdinand Möbius and Johann Benedict Listing “discovered" in 1858.

They discovered its curious properties, and started—as mathematicians do—investigating them mathematically. In this case, that will need instructions mathematically encoded—like “give one end of the strip a twist". Which is how mathematicians come up with