Let's continue our project of exploring the geometry of flat surfaces by living on them. If instead of identifying opposite sides of a square directly, we reverse direction in one of the pairs, we get a new surface, called a Klein bottle.
First, a necessary and fun diversion. In general we'll deal with surfaces which have no edges or boundaries, with one exception: the Möbius band. Identify two edges of a square with opposite orientations (or two short edges of a rectangle, if you want to actually make it with paper), and you'll have a space with a boundary called a Möbius band.
A Klein bottle is a surface with no boundary but with the same property as you've observed above in the Möbius band: going in some directions returns you to the place you've started from but mirror reflected.
Check your answer to the above activity here. Just as in the case of the torus, the "seams" or edges of the square are not actually there. They are not perceived by the inhabitants of our spaces, they simply help us define the geometry of these spaces. From personal experience, I think this fact is a lot harder to accept when it comes to the Klein bottle; there's a natural tendency to picture a magical boundary that flips you over when you pass through. As I hope you've seen in the past exercise, this is not so. This fact is so important in our further study of geometry, that I'll say it again:
Having said that, as inhabitants of a Klein bottle, we can in fact draw a fundamental polygon, which is just math-speak for the square containing the whole area of the Klein bottle, like the ones we've been drawing to define the Klein bottle to begin with. Here's one way to do that.
Here's a pictorial illustration.
So far we've identified the sides of a square in two ways, either gluing opposite edges directly and getting a torus or reversing the direction of one of the pairs and getting a Klein bottle. To the right you see another identification scheme.
As you've discovered above, I hope, this space has a different geometry from the torus and Klein bottle. Both of those had a flat geometry: every point looked like a point in the plane, even the corner point of the fundamental polygon. This space does not look the same at every point; it has four points with an angle deficit, as is illustrated below.
Points such as the corners of the "sphere" square above are called cone points. Their angle does not add up to 360 degrees, so they can't lie on a plane. Here comes another subtle but very important point: the geometry of the identified square above and its folded version, the double cone, is not the same as the geometry of the sphere we've explored in the last section. Your avatar the cyclops can easily tell the difference between the double cone and the sphere by looking around. However, topologically the double cone and the sphere are the same. If the distinction is not clear right now, do not despair! We'll talk more about geometry and topology in just a little while. For now, on to a new surface!