|One of the things mathematicians like to do with their objects to make representations of them. Sometimes these are purely symbolic; sometimes they’re diagrams or drawings. We like to have lots of different ways to represent them because each one shows us different aspects of the object and makes different kinds of manipulation easy or hard.
In our first Strange Spaces class we made some 2-manifolds out of paper. At this stage the subject is very new and hard to grasp, so this exercise is about more than just getting to know these objects: it’s also about beginning to scope out some different ways of looking at them.
The most interesting one that’s easy to make is the Möbius strip. This is made by taking a strip of paper and sticking the thin ends together as if you were making a cylinder, but first twisting them (source):
Perhaps you know that cutting a Möbius strip up produces unexpected results:
This behaviour is quite famous, and it’s usually presented as a party trick that’s amazing and mysterious. As mathematicians we want to do a bit better than just being amazed, though. So one question we might ask is: How can we get a better understanding of this space so that we can predict what will happen when we cut it in different ways?
It’s possible to predict what will happen by drawing on the model itself, as this diagram shows (source), but it’s quite easy to make a mistake and not so easy to go from the object in  to the mental image in :
It would be even harder to picture this in your head without any physical model, and often with high-dimensional or weirdly-connected spaces we don’t have that luxury — we can’t even make a sketch like this one to help us. So this kind of representation is of limited use.
Here’s a completely different representation — this time a symbolic one (source):
Now, this has its uses too; for example, I used something very similar to make this image and do some calculations with the curvature (geeks may like to know that the black arrows are the standard basis of the tangent space at the point where they meet):
The symbolic representation is really useful for this sort of thing but it’s no help at all in figuring out what happens when you cut the Möbius strip up, and why you get two linked strips with different numbers of twists. In fact it’s hard to tell just by looking at the equation that there’s any twist in the original strip!
In class we worked with a different kind of representation: a polygonal presentation. This is diagrammatic, and it emphasizes how the Möbius strip is made by gluing together the edges of a rectangle with a twist (source):
The diagram means “take a square and glue together the sides labelled A in such a way that the arrows match”.
This may not seem like much of a representation. It doesn’t look like a Möbius strip, and nor does it seem to have any mathematics in it that we can manipulate. But what it does is throw away all the information about the situation except what we’re interested in: how the space is connected to itself. It turns out (as we discovered in class) that we can explain the weird behaviour of cutting up the Möbius strip rather easily by cutting up this diagram. In future classes we’ll do more with these diagrams and they’ll help us find our way around surfaces like the Klein bottle that can’t possibly exist in our 3D space.
The point here is really something familiar to us all: there can be many ways to represent the same object, each of which tends to make certain aspects of it more readily available to us. Picking the “right” representation isn’t a matter of correctness so much as “seeing things in a way that helps us answer our questions”. And that will be one of the themes of the whole course.