Teacher resources and professional development across the curriculum
Teacher professional development and classroom resources across the curriculum
Let's go back and check in with our adventurous Flatland explorer. Having completed her experiment with the red and blue threads, she decides to set out once more, this time in a southeasterward direction. She travels a fair distance and realizes that she has not seen either her blue or red thread anywhere. Making a mental note of this, she treks on until she sees a building in the distance. As she approaches it, she notices something strange. Although the building appears to be her house, it has some odd features. The address numbers are reversed, as if they were written in a mirror, and upon further observation, she realizes that her entire house has been reversed. The tree that used to be to the left as she approached her front door now is on the right. Her bedroom, which used to be the last door on the right of her hallway, is now the last door on the left.
Suspecting some kind of practical joke, she seeks out her neighbors to help get to the bottom of this. When her neighbors see her, they are shocked at her unusual appearance. All Flatlanders have their eyes to the north of their mouths when facing west and their mouths to the north of their eyes when facing east. The orientation on our explorer's face is the opposite. Her mouth is above her eye when she faces west, and her eye is above her mouth when she faces east.
Something happened to our explorer on her latest journey that reversed her orientation, relative to how she started out. This is why everything appeared as a mirror image to her. This strange part of Flatland was hitherto unknown, and it is hard for the average Flatlander to figure out what happened.
As three-dimensional observers with the advantage of an extrinsic view, we have the perspective to find a somewhat more satisfactory explanation. The region that our explorer experienced is a place where orientation is meaningless. This means that if a Flatlander takes a trip through this region, they will return "mirrored," as our explorer did. A surface with this mirroring characteristic is known as a Möbius strip.
We can see that this surface has only one edge, and that while it appears to have two sides, it really has only one. To create a model of this surface, simply take a strip of paper, put one twist in it, and then attach the ends together. Tracing the surface with your finger will convince you that both sides of this object actually are one and the same. When our Flatlander explorer took a trip through "Möbius land," completing one cycle of a Möbius strip, she returned to the point where she began, reversed in orientation. (Be careful—remember that a Flatlander lives "in" the surface and not "on" the surface.) This kind of surface, in which paths exist that can reverse one's orientation, is known as a non-orientable surface.
The Möbius strip is not the only kind of non-orientable surface. Another well-known example is the Klein bottle, shown here intrinsically.
Notice that following some paths on the Klein bottle will reverse one's orientation and following others will not. For instance, in the following diagram, east-west paths are reversing, whereas north-south paths are not.
This surface is a bit stranger than a Möbius strip. We can think of a Klein bottle as a surface whose inner face and outer face are the same. To take an extrinsic view of this surface would require that we somehow embed it into 3-space.
Unfortunately, just as we found that certain graphs cannot be embedded in the plane without intersecting themselves, the Klein bottle cannot be embedded in 3-space without a self-intersection. We can, however, create what is called an "immersion," and one possible immersion looks like this:
Mentally walking along the surface of this object should convince you that its "inside" is the same as its "outside."
Both the Möbius strip and the Klein bottle are relatively easy to picture extrinsically, but the third non-orientable surface that we shall investigate is really best understood intrinsically. It is known as the "real projective plane" or, more commonly, just the "projective plane." Intrinsically it looks like this:
Notice that a Flatlander traveling across this surface would be reversed no matter which path she chose to follow.
An interesting aspect of the projective plane is that it can be used to construct other surfaces. In fact, by cleverly attaching two projective planes together, we get a Klein bottle. To perform this "operation," we first take two projective planes and unhinge them at one connection:
We then connect them together to form the square, with a diagonal representing the seam. Now, if we rotate the planes with respect to each other, we end up with a diagram that resembles a Klein bottle with a diagonal. Because the diagonal is interior to the shape, we can disregard it, and—voilà!—we have a Klein bottle.
This process of combining two surfaces to create a third surface, possibly of another type, is a powerful idea. It helps to explain the structure that our Flatland explorer found. Although she found her world to be like a torus in most respects, it included a region that reverses people. We can think of this as equivalent to the surface of a torus glued to either a Möbius strip, Klein bottle, or projective plane. In fact, it is possible to add together all types of surfaces to create new ones. In the next section we shall see how this concept leads to a new type of algebra, in which we use surfaces instead of numbers.