In the last post, we introduced the idea of manifolds. A two-dimensional manifold is essentially a shape on which we can go in two directions, say up-down and left-right, but not forward-backward as well. We talked about some simple examples, like a sphere or a torus, but there are so many more wonderful manifolds than that!
Let me give you a brief tour of the manifold menagerie.
The first manifold I want to show you is an interesting variation of the torus. Last time, we showed how you can make a torus by considering a rectangle, and identifying opposite sides.
Now, notice how the arrows are pointing in the same direction. What would happen if we changed the direction of one of the arrows? You’d get what we call a Klein bottle.^{1}
What this means is that the left edge is still identified with the right edge, but with, for instance, points near the bottom left being identified with points near the top right edge.
Another way to visualize this is that if you played Asteroids on a Klein bottle, and you flew through the left side facing up, you would come out the right side, facing down!
What does this Klein bottle thing even look like, anyway?
Well, first we can roll up the rectangle, like we did with the torus. But we can’t bend the two ends together quite like before because the orientation (i.e., the direction of the arrows) doesn’t match up. Instead, we have to do something funny.
Take the end of the tube through the wall near the other end, then bend it up through the end. Then, flange over that end to match up the two ends. Okay… that description needs a picture.
Weird looking, right?
Now, you may want to raise an objection. Dr. Dilts, you may say, isn’t that not a manifold? After all, where the tube intersects itself, it doesn’t look like !
You’d be right to mention that. In fact, that’s one of the interesting things about this example.
If you look back at the rectangle representation of the Klein bottle, there’s nothing that stops it from being a manifold. There’s no self-intersection there. The problem is not in the Klein bottle itself, but in how we chose to visualize it in three-dimensional space, !
As an anology, consider a circle.^{2} We can think of it as a loop of string. In that standard configuration, it looks like a manifold, since if you zoom in, it looks like a line, i.e., a one-dimensional space. But we could pick up the end and pull it over itself like such:
It no longer looks like a manifold, but, again if we think of it as a string, the places where the loop overlaps itself are not the same points; they just look like it because they overlap. The crossing lines don’t “see” each other.
Again, the problem is not with our loop of string, which certainly is a manifold. The problem is with how we are visualizing it in the plane, .
So, the Klein bottle is a manifold.
This raises a question. Can we visualize the Klein bottle so that it doesn’t intersect itself?^{3}
The answer is yes, but, surprisingly, you can’t do it in . You need a fourth dimension to do it!
The idea is something like this.
If we go back to the loop of string example, where the string overlaps itself, the visualization intersects itself. However, it’s easy to resolve this problem. Simply pick up the string near where the intersection is; in other words, pull it from two dimensions into the third.
In order to visualize the Klein bottle without an intersection, we have to do the same thing, only with a fourth dimension. You grab the Klein bottle near the intersection, and pull it from three dimensions into the fourth.
Weird, huh?
Now, this worked for the Klein bottle, this trick of pulling it into a higher dimension. But you can imagine there are other manifolds that have this same problem of self-intersection. (In fact, we’ll even get to another one in this post!)
If you pick any manifold, is there always a visualization that doesn’t have self-intersections?
The answer is yes, though it’s hardly obvious how to prove such a statement. The Whitney embedding theorem says that any manifold can be visualized without self-intersection, but it may require a high dimension to do so. Precisely, it may take as many as twice as many dimensions as the manifold has.
If the manifold has dimensions, this visualization may require as high as dimensions. Our Klein bottle (with ) is an example of this, as it requires dimensions. A six-dimensional manifold might take twelve dimensions!
Okay, back to the funky Klein bottle thing.
Guess what? It doesn’t have an inside or an outside! Weird, yes? This fun fact about the Klein bottle is exactly like how a Möbius strip has no front or back.^{4}
With a sphere, if you think of yourself walking on the outside surface, you can’t “walk” to the inside of the surface; you’d have to punch a hole through. But look closely at the Klein bottle, if you started walking on the “outside,” and followed along just like you know you should, you can end up underneath where you began.
Trippy.
Okay, the Klein bottle is weird.^{5}
Let’s talk about some more pedestrian^{6} examples.
In the last post, we talked about the sphere, by which I mean a shape like the surface of the Earth. We call this the two-dimensional sphere, (read “S 2”), since there are two directions you can go.
Along with that, you have the one-dimensional sphere, . This has one direction you can go… in other words, it’s a circle!
We can go higher in dimensions as well. What is , the three-dimensional sphere?
Well, it’s not the inside of the sphere, which is what your mind might go to first. Instead, it will help to use analogy.
If you look at , horizontal slices of it are just circles, .
So, as we take slices of , we have small circles that get larger till we pass the equator, then get smaller again till they disappear.
We can visualize the same way. The problem is, it’s naturally in instead of , so it’s harder to see. To help us, let’s think of as the three normal dimensions, plus time.
So, if we take , and we look at what we can see at any instant in time (which is equivalent to looking at horizontal slices in the previous example), we can see a sphere, , which starts small, grows till it reaches the “equator,” then shrinks back down till it disappears.
Yeah, four dimensions is hard.
MORE manifolds!
The torus we talked about before is just one example of a torus. Since it’s a two-dimensional manifold, we call it (read “T 2”). Using the same idea, it’s not hard to make higher dimensional versions.
If we take a cube (instead of a rectangle), we can identify opposite sides of the cube. So, if you are playing (3-d!) Asteroids in this, and you fly out the top, you’d fly in from the bottom. This gives us a new, higher dimensional torus, .
Pretty cool. Of course, it’s much harder to visualize as sitting inside of . Good thing we don’t have to!
There’s another variation we can do.
Instead of taking a square, take an octagon. Now, identify the sides in this pattern.
Now, it’s much harder to see what’s going on without a hint. What is this shape?
It’s a two-holed torus.
You can actually do this for any number of holes. You just need 4 sides for each hole you want your torus to have.
Let’s see, what else weird can we do?
Oh, I know!
Okay. Take the sphere (). Now, for each point, identify it with the point on the exact opposite end of the sphere.
Another way to think of this is to think just of the upper hemisphere of the sphere. Then, just on the equator, again identify the points on the opposite side.
In other words, if you fly south, and go past the equator, you end up coming north out of the equator on the other side!
At first consideration, this really seems to just be a sphere, since on a sphere, you can fly through the south pole just the same. But it really is different. It’s called (read “R P 2”), where the stand for “real projective” space.
And it’s really weird.
To explain it, we’re going to need one of those funny mathematical objects; an infinitely stretchable and infinitely compressible rubber band.
Look at the sphere. If you put a rubber band on it through the north and south poles, you can always have the rubber band shrink down, like this:
However, if we took the equivalent path on , it can’t shrink down! See, if you try to move the parts that hit the equator closer to each other, you can’t, since they, by the definition of the space, have to stay on the opposite sides of the hemisphere! Essentially, you can only rotate the rubber band, but never have it come off.
But, if we wrap the rubber band around twice, all of a sudden we can shrink it down.
Wait, what?
The trick is to run one part down each side of the hemisphere. Thus we can have two small half circles getting smaller, and finally shrink to nothing like on a sphere.
So, in , one closed path is stuck. But if you wrap it around twice, all of a sudden you can shrink it down to nothing!^{7}
So this new space is not a sphere!
Again, we call this space . As you can see, it turns out to be about as interesting as the Klein bottle. In fact, we can even use a similar rectangle to define it, like this!
Like the Klein bottle, cannot be visualized in three dimensions without self-intersections. Here’s one way to look at it, with those opposite parts of the equator matched up.
also doesn’t have an inside or an outside. Which is really weird, since it seems like if you take a sphere, anything you get from it should still have an inside… But, you know… strange but true statements like this are the heart of mathematics.
There are plenty of other weird manifolds out there, but that’s a good introduction to some interesting and important ones.
Next, we want to come up with a way to measure distances on a manifold, without visualizing it in a space like .
- Apparently, it was originally called Kleinsche Fläche (Klein surface), but, perhaps due to its appearance, it was misinterpreted as Kleinsche Flasche (Klein bottle), which may have led to the use of this term in German as well. ↩
- Remember, a circle is the outer edge, not the filled-in shape. ↩
- A visualization of a manifold (in , for instance, and with a few other conditions) is called an immersion. If the immersion doesn’t intersect itself, it’s called an embedding. So, our current visualization of the Klein bottle is an immersion, and the question is whether we can find an embedding. ↩
- In fact, a Klein bottle is two Möbius strips with their edges glued together. ↩
- Klein bottles are so cool that you can get blown-glass Klein bottles. The man who runs that site, Cliff Stoll, looks like a mad scientist. And he’s obsessed… Check out this interview of Cliff Stoll on Numberphile. He also shows off his Klein bottle hats and Möbius scarfs! ↩
- Pun! Ba-dum-ching! ↩
- This idea of drawing paths and asking whether or not they can be shrunk to nothing is an important one in topology, called the fundamental group of the manifold. I was about to link to the Wikipedia page, but on second glance it’d be a horrible introduction… ↩