Real Mathematics – Strange Worlds #5

WHERE AM I?

Let’s assume that you have an object shaped like a sphere. Take a pen and draw a starting point on the object. Then, select a direction and move in that direction from your point. When your tour is completed, you’ll be ending up at your starting point in your original direction.

This is an obvious fact which would never change no matter which starting point or direction you choose.

Orientability: Mathematical definition of orientability is really complicated, even for a mathematics graduate. Which is why I will use an example to define it: Imagine that you are travelling to east from Hamburg with a zeppelin. If you continue travelling, and if you are lucky enough to survive, you would eventually end up at Hamburg while your zeppelin faces east. (Please leave you flat-earthers) Actually, it doesn’t matter which direction or city you choose, result will always be the same. If an object or a surface has this specialty, then we say that it has orientability.

It is absurd to think that you finish travelling as a mirror image of your starting state. If you move to the east from a point, you won’t accidently end up at the same point with your direction flipped to west. In other words, you won’t finish taking a tour on a sphere as your mirror image.

Two types of mirror images. Obviously I am talking about the one on the right.

Is it possible to have an object that doesn’t have orientability? Is it possible to go east and when you arrive at your starting point you realize that you are standing upside down?

YES!

Möbius Strip

Cut a rectangle shaped paper. This paper has two faces.

There is no interchanging from one face to another. It means, if you start moving on one face, no matter where or in which direction you started, you won’t end up in the other face. We can call these faces “roads”. So a paper has two different roads.

A paper can have two faces (roads). You can’t go from I to II.

Now glue the ends of this paper together. You will have a cylinder. A cylinder has two faces like a flat paper. It means that a cylinder has two different roads. It is not possible to use one road and end up in the other road. This is why a cylinder has orientability.

Use a transparent paper and make a cylinder. Mark a starting point on your cylinder and move it around your finger until you are at your starting point. You would end up in the same conditions.

Let’s go back: Take your flat rectangle paper and while bringing their ends together twist one end 180 degrees. Mathematicians call this shape a Möbius strip.

While constructing, that 180 degree twist on one end is the only difference between a cylinder and a Möbius strip. But, this difference has absolutely mesmerizing results. First thing to notice is that there is only one face in a Möbius strip. That means a Möbius strip has only one road.

Let’s use a transparent paper again and construct a Möbius strip. Select a starting point and move the strip around your finger. When you end up at your starting point, you’ll see that “up” became “down”, “down” became “up”. In other words, you ended up as your mirror image on a Möbius strip.

Up-right became up-left as if there is a mirror between them.

Then, there is no orientability for Möbius strips. Also, if you make a second tour on it, you would end up at the starting point as your original state.

A little bit of history

It is surprising to learn that Möbius strip was first discovered around 160 years ago. This simple but mysterious shape took its name from the German mathematician August Möbius. Although, Johann Listing, who is another German mathematician, was the first person who published about Möbius strip.

August Möbius (left) and Johann Listing (right).

That is because August Möbius and Johann Listing discovered about Möbius strip independently almost at the same time. After looking through their personal notes we understood that August Möbius discovered the strip 2 months before Listing did. Even though Listing was the first person to use the word “topology”, 2 months gave August Möbius a better mortality.

One wonders…

1. Construct two Möbius strips. While twisting them, twist one of them to the right and the other one to the left 180 degrees. Do you see any difference?
2. Construct a cylinder and cut it into its middle in the direction parallel to its longest edge. You will end up with two cylinders which are little replicas of the original cylinder.
   Try the same thing on a Möbius strip. What is the result? Why did you have that result?

M. Serkan Kalaycıoğlu