A Very Interesting Shape

Nov 19, 2015 | Sharyland




(Image: Möbius Strip II (Red Ants), by M.C. Escher, 1964, via Flickr)

November 17 marks the birthday of German mathematician August Ferdinand Möbius (November 17, 1790 – September 26, 1868), which means it’s the perfect time to wax poetic about one of the coolest things you can make out of a strip of paper—the Möbius strip, discovered by Möbius in 1858!

You can make your own Möbius strip by cutting out a strip of paper, giving it a half twist, and sticking both ends together. It’s simple by design, but this shape has some interesting traits.

While it may appear otherwise, the Möbius strip has only one side and one edge—in math terms, we describe it as non-orientable. Though the piece of paper from which it was created is orientable and very clearly has two sides, once you turn it into a Möbius strip, it becomes a one-sided object. Crazy, huh?

You can test the Möbius strip’s one-sidedness yourself. Take a pencil and draw a line along the strip’s surface—while you may have to wiggle the strip around a little to get this right, it’s possible for your pencil to travel along the entire strip and return to its starting point without you lifting it off the paper.

Now try cutting the Möbius strip lengthwise down its midline. Surprise—you’ll end up with a twisted strip that’s twice as long! Next, try cutting this longer strip lengthwise down its midline. The results may surprise you!

So, why did the chicken cross the Möbius strip? To get to the same side.