Thanks to plf515 for letting me host math mania this week. Today I'd like to talk about manifolds.
A manifold is the shape of an imaginary universe. This isn't supposed to make sense yet, it's only meant to give you an idea of what we're trying to do by studying manifolds: we're trying to understand the shapes of imaginary universes.
This series is for anyone. There will be no advanced math used. Nothing beyond high school, usually not beyond grade school. But it'll go places you didn't go in elementary school or high school.
If you "hate math" please read on.
If you love math, please read on.
I welcome thoughts, ideas, or what-have-you. If anyone would like to write a diary in this series, that's cool too. Just ask me. Or if you want to co-write with me, that's fine.
The rules: Any math that is required beyond arithmetic and very elementary algebra will be explained. Anything much beyond that will be VERY CAREFULLY EXPLAINED.
Anyone can feel free to help me explain, but NO TALKING DOWN TO PEOPLE. I'll hide rate anything insulting, but I promise to be generous with the mojo otherwise.
The mathematics of squishy shapes
I'll tell you what I mean by universe, but first I'll tell you what I mean by shape. Manifolds are a concept from topology, and topology is a kind of math. In topology we study shapes, but we imagine that every shape is made of very squishy rubber. Circles become rubber bands, and spheres become malleable, hollow rubber balls.
This is very nice if you're a qualitative thinker, rather than a quantitative thinker. If numbers aren't your favorite thing then this could be the math for you. We don't have to think about the diameter of a circle, or the area inside, or whether it's really a circle or an ellipse, because circles of all shapes and sizes are the same to us. They all look like squishy rubber bands.
Up close, universes all look the same
Now, what do I mean by universe? Let's start by examining our own. One thing I notice about this universe is that it appears to be three dimensional. This means that there are essentially only three directions we can move: up-down, left-right, and forward-back. Of course we can also go diagonally, we could go kind of uppish-leftish-forwardish, but this isn't really a new direction so much as a mix of the other three.
Another thing I notice about this universe is that it always looks the same no matter where I go. Of course the colors change, the furniture changes, the scenery changes, the company changes. But the shape of space itself doesn't change. Whether I'm in my room, in the ocean, at a concert, in an airplane, I can always imagine a ball of space around me.
This is the essential characteristic of a universe: universes always look the same to their inhabitants. Space always looks the same up close. This is so important that we take it to be our definition: A three-dimensional manifold is a shape that always looks like a ball, up close. In other words, if you pick any random point on the shape you can always cut out a ball of space around it.
At first the only obvious three-manifold is our own universe, but we'd like to imagine new ones.
Two-dimensional universes
How can we invent new universes? For practice let's start a dimension down, by imagining a two-dimensional person. This person is flat like a piece of paper, and can only move left and right and up and down, but never forward or backward. This flat person's natural habitat is a flat universe, like a plane or screen. But this isn't the only option; we could also imagine this person living on a sphere.
When I say sphere I mean a round hollow shell, like a balloon or a hollow globe. Since our flat people are made of rubber, we can mold them onto this globe, and let them move around on it. They can move north and south, east and west, the globe the only reality they know.
While a sphere seems three-dimensional to us as we look at it from the outside, our flat people are none the wiser. Up close, a sphere looks flat, and it always looks the same. So we consider both planes and spheres to be two-dimensional manifolds, since they're both good universes for flat people.
The surface of a donut, called a torus, is also a two-dimensional manifold. A torus is not a donut, it's only the chocolate-frosted outside of one. And since we're assuming it's made of rubber, we can picture a torus as a rubber bicycle inner tube, without the valve.
We can imagine molding our flat people so that they fit on the surface of an inner tube, and then setting them free to live their lives there. We could watch them move through the hole, around the hole, blissfully unaware of the shape of their universe. And if we pick any random point on that inner tube, we can always cut out a disk of rubber around it. This shape, like the sphere, looks flat up close.
The Pac-Man approach to building new universes
Here's another approach: Consider Pac-Man, the video game character. He's a flat person living in a flat world, and at first his world seems to be a rectangular screen. But there's more to this rectangle than meets the eye.
When Pac-Man leaves through the right edge of the screen, he reappears at the same height on the left edge. When he passes through the top edge he reappears at the same place on the bottom edge. This tells us that, as far as Pac-Man's concerned, the left and right edge are the same, so we might as well glue them together. Picture taking a rubber rectangle and bending it so as to bring the left and right edges together: You'll find yourself holding a rubber tube.
Now Pac-Man can also go straight from the top edge to the bottom edge, so we might as well glue the top and bottom edges together as well. Picture yourself holding this rubber tube and bending it so as to bring the top and bottom edges together, and you'll find yourself holding a torus. This shows that Pac-Man's universe is actually a torus.
We can take the Pac-Man approach to build new three-dimensional manifolds. Imagine that you are like Pac-Man, except that instead of living in a rectangular screen you live in a box-shaped room. This room has an east wall, a west wall, a north wall, a south wall, a floor, and a ceiling.
But it's a very special room -- so special that you can walk (or float) through the walls. But when you walk through the west wall, you reappear at the same place on the east wall. And when you walk through the north wall, you reappear at the same place on the south wall. And when you float through the floor, you reappear at the same place on the ceiling.
This defines a three-dimensional manifold. This room is a self-contained universe that you can never escape from. We call this world a three-dimensional torus, because its definition is so similar to the Pac-man definition of the usual torus.
But what does this world look like? We know what it's like to live there – it's like living in this magic room where you can walk through walls. But what does this manifold look like from the outside? We could try building it. You could imagine taking a block of squishy rubber to represent the room, and bending it to bring the east and west walls together. And if your rubber is squishy enough you could also bend it again to bring the north and south walls together. But if you then tried to glue the floor to the ceiling, you would run into some problems. It's not because you're not artistic enough, it's because this shape cannot exist inside our universe. There just isn't enough room for it in this reality, so you can never build it in your room. You have to build it in your mind, instead.
The people have the power to visualize four-dimensional space
There are many ways to imagine the three-dimensional torus, and one is to visualize yourself in four-dimensional space, with this strange shape standing next to you. You might say to me, "But Gauss Bonnet, we can't possibly visualize four-dimensional space." And to that I say... Yes! We! Can!
Visualizing higher dimensions is difficult. It takes practice. But it can be done, and there's no one right way of doing it. Every person reading this diary might come up with a different way of imagining four-dimensional space, and that's wonderful. This is the difference between humans doing math and a machine doing math. People are creative, we're intuitive, and we're different from each other. This is why everyone has something to contribute, and why it's so important that the mathematical community be open, that it welcome people in, that it not keep people out.
This brings me to my questions for you. (You might call them assignments, but don't worry, they're optional.) My first question is: How do you personally think about four-dimensional space? Or where do you get stuck in trying to think about it?
What other manifolds can you imagine? You might start in dimension two, and use the Pac-Man technique. What would happen if you glued Pac-Man's screen together in a different way, or if he lived on a screen with a different number of sides?
The spheres we talked about are actually called 2-dimensional spheres, because they make good homes for two-dimensional people. What do you think a 3-dimensional sphere could be?
Any guesses what the phrase "4-dimensional manifold" might mean?
And what are your questions for me?