How to wrap and twist a cylinder into a Mobius strip?

Question

In this article, the author said that a cylinder, wrapped twice on itself, with a twist, to form a Möbius strip.

I wonder:

1. What does they mean by a cylinder wrapped twice on itself?

2. From that how can they twist it into a Möbius strip?

I can't imagine the process. Please explain for me.

Answer 1

Start with a 2-dimensional strip of length 2L and width W: $[0,2L]\times [0,W]$. In the article you identify this strip with a music sheet. If you visit it from 0 to $2L$, you are playing a total of $2L$ time of music, then you stop.

If you bend it and you identify $(0,v)$ with $(2L,v)$ for $0\leq v\leq W$, you get a cylinder. If you play it starting from $t=0$ you eventually reach $t=2L$, from which your score starts looping, and you repeat exactly the same music again and again every $2L$ time.

Now you create the same effect with a different realization. Start again from the strip. You bend half of it (from 0 to $L$), then you twist it once, and you identify $(0,v)$ with $(L,W-v)$ for $0\leq v\leq W$. Now you created, with half of the strip, a Möbius band. With the remaining part, you just continue the identification in the obvious way.

Namely, you follow the previously created Möbius band, sticking the portion at $(L+t,v)$ to the one at $(t,W-v)$ for $0\leq t\leq L$ and $0\leq v\leq W$. You finish the process by identifying $(2L,v)$ to $(L,W-v)$, and so also to $(0,v)$ as before, for $0\leq v\leq W$. If you play your music along the Möbius band, then you are playing "simultaneously" the music at time $t$ and the flipped version of the one at $t+L (mod 2L)$, or viceversa (after $L$ time.

Indeed, you agree to play what you see on one side with the local orientation transported from the initial one to that moment. But in the example of the article the music score has a symmetry that makes the music at time $t$ actually coincide with the flipped version of that at time $t+L (mod 2L)$. So first you play $L$ time of music, then you play it again, but flipped, and then at time $2L$ you exactly loop back.

As you see, your cyclinder wrapped "on itself", and it did it "twice". And between the first and second visit of "itself" it "twisted" once (flipping twice does the effect of a cylinder again, as opposed to the effect of a Möbius band).

In other words, you have a discrete action of a group $G$ (here $\mathbb Z/2\mathbb Z$, corresponding to youreflection symmetry), and you have a map between your original cylinder $X$ presented music and its quotient "wrapped up cylinder" $X/G$. This map is exactly a double covering map, as André Henriquez said.

Notice that in the photos in the article, the cylinder is 1-layer thick, while the Möbius band is slightly noticeably 2-layers thick.

Answer 2

Sorry for the double answer, but I noticed that your question can be read from a different perspective, and this admits a completely different answer.

In the other answer I described how a rectangular strip can be wrapped on itself twice, with a twist, so that it forms a Möbius band, and that can be seen as a degree 2 quotient of a cylinder.

If you ask, on the contrary, how can you wrap twice directly from the cylinder version to the Möbius one, without passing from the open strip, then the answer is that you can't.

At least, not in 3D, without cutting moves, and starting and ending with standard 3D presentations. Indeed the two edges of the original cylinder form two unlinked simple knots, while its two edges in the final wrapped form ($\mathbb Z/2\mathbb Z$-bundle over the Möbius band) are 2 simple knots with linking number $\pm 2\neq 0$.

Thus, if you want to visualize it concretely, (other than just doing the math) these are the instructions: take a long and narrow paper strip. On one side only, color in blue one lateral edge, and in red the other. Twist the strip four times. Identify the extremities so that the colors match on the same side, so you realized a (4 times!) twisted cylinder in space, a version that actually can be wrapped onto itself twice.

To make the double wrap, take the portion where you joined the extremities, and the "opposite" one, located between the second and the third twist of the strip. Make them close to each other, both with the colored side facing you. With an able rigid motion (in the correct direction!) identify the two portions, making sure that

• the non-colored sides touch each other, while the colored ones point outward (one leftward, the other rightward);
• the blue edge of one portion-side correspond tho the red one of the other;
• the rest of the strip behaves nicely (this is because there are two natural rigid motions that permit the previous point; one of them works, while rotating in the opposite direction makes a mess)

Adjust the lengths and enjoy your creation.