Non-orientability 1: Möbius strips

So. Non-orientability. It’s a long word. What on earth does it mean?

Let’s start with a word that we do know. Orient means “to determine the position of, in relation to the points of the compass”. To be oriented means that you know, for example, which way is north. But that’s quite an Earth-specific definition. If you were floating in space, how would you define ‘north’? Perhaps you could use the concepts ‘left’ and ‘right’, ‘clockwise’ and ‘anticlockwise’ to give someone directions. These words would always work, right?

Well, be prepared to have your mind blown! Mathematicians have discovered universes in which the concepts of ‘left’ and ‘right’ make no sense at all.

The simplest example of such a ‘non-orientable’ space is called a Möbius strip, and is constructed as follows. Take a long thin strip of paper and connect up the opposite ends with a single half-twist. You should have something that looks like this:

Möbius strip

Möbius strip

The first interesting thing to note about this object is that it has only one side. On your piece of paper, take a pen and trace a line along the length of the strip; you will soon see that you arrive back where you started but on the other side of the paper. So we have already lost the concept of a ‘front’ and ‘back’. What about left and right?

Meet Flib. He is a 2-dimensional creature whose universe is a Möbius strip. Due to an unfortunate paint accident, he has to live with the fact that his right hand is blue. One day, Flib is feeling adventurous and decides to explore the rest of his universe. Dum de diddly dee, along and along he wanders, until eventually he finds himself back where he started. But what is this? His friends all notice that now his left hand is painted blue instead of the right one, whilst poor Flib is wondering why his friends have all turned into the mirror images of the ones he left behind!Flib on a Möbius strip

This little story shows us that the notions of left and right are simply not well-defined on a Möbius strip. And the Möbius strip is not the only example of a non-orientable space. As the weeks pass, I will show you other examples (sometimes in higher dimensions!) and will ask the all important question: is our own universe orientable?


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