Most people know that if you take a long piece of paper, twist one end 180 degrees and then glue it to the other one, you get a shape with only one side and edge. Indeed, if you have never seen this, try it out right know! You can place your finger on one side or edge and follow it around until you end up on the “other” one. We call this shape a Möbius strip and would like to study it mathematically. To do so, we must abstract away the physical paper. From this, we will construct a Klein bottle, for which the “inside” and “outside” are also the same.

Surfaces

Sometimes drawing pictures is a valid method for constructing spaces, like the method of surfaces in topology. To abstract away the physical limitations of real paper, we just draw the piece of paper as a polygon and indicate which sides should be glued together. Edges with the same letter are identified in such a way that the arrows match. So this is a Möbius strip:

We start with a piece of paper, take two opposite edges and glue them together with a rotation of 180 degrees. We assume that this abstract paper is infinitely stretchy and sometimes even allowed to pass through itself. This is because in topology, we consider spaces only up to deformation. The latter is mainly needed in three dimension because some surfaces cannot exist otherwise.

Since we said that edges with the same labels should be glued together, we can also introduce new cuts and even split the piece into multiple as long as we keep track of how we cut. Similarly, we can identify the edges with the same label. So the following is still a Möbius strip and we see that it has only one edge, namely the one at the bottom:

We can also construct other interesting spaces this way. For example, if we do not rotate one side of our paper, we get a cylinder (left). If we also glue the other two edges together in a certain way, we get a donut (middle). We can also relax what a polygon is and see that the shape on the right is a sphere.

One nice thing about surfaces is that you can always recreate them using real paper (although the torus and sphere might pose some difficulties). So if you do not believe me, you can get a piece of paper and glue it accordingly to check.

A Möbius strip appears every time one wants to consider unordered pairs on the cylinder. We can parametrize the cylinder above by the unit square and divide by the equivalence relation generated by $(a,b)\sim(b,a)$. This corresponds to “folding” the representation of the cylinder along the main diagonal. Doing this results in the triangle representation of the Möbius strip we saw before.

Non-orientability

In its real-life version, we have seen that a Möbius strip has only one side. Since we cannot define an “inside” or “outside”, Topologists call this shape non-orientable. (Note that for example this does not hold for a cylinder.) We can already see this using the tools we have so far. Suppose we had a little loop with orientation living on the Möbius strip. Just by moving around, it can reverse its direction:

This is a direct consequence of the twist in the paper. This cannot happen if we can define a clear inside or outside on the shape. In that case the loop can only stay on that side and moving around would never change its orientation since it is essentially moving on something that looks like $\mathbb R^2$. You can try with a cylinder to see this for example.

Gluing two Möbius strips together

As we have already seen, a Möbius strip has only one edge. Therefore, there is a canonical way of gluing two Möbius strips together, namely along this edge:

You might notice that we could have also flipped the bottom one before gluing them together. You can easily check that we can use a similar construction get the same result. At first, this does not look like much, but like before, we can modify it to get:

This is the standard representation of a Klein bottle! We can again glue the edges with label $b$ together to get a cylinder. Now we would like to glue the other two edges. But we cannot do it the same way as we did with the torus, because then the orientations do not match. Instead we need to somehow reverse the orientation of one of them. We can do this by pushing one end of the cylinder through one of the walls and pulling it through the inside to match the orientation. If you are unfamiliar with this, an animation might help: https://www.youtube.com/watch?v=E8rifKlq5hc . So we have shown that gluing two Möbius strips along their only edges gives indeed a Klein bottle! Using the same trick as with the Möbius strip, we can also see that a Klein bottle is non-orientable, explaining the meme at the beginning.