Have you ever played the game Asteroids?
In this game, you fly a little triangular spaceship and try to destroy all of the asteroids floating around you while avoiding collision.^{1} One interesting feature of this game is that if you fly through the right side of the screen, you come out the left side. If you fly through the top of the screen, you come out the bottom.
In other words, the game takes place on the surface of a donut.
This visualization is hard the first time you think of it, so let’s work through this.
We can think of the game area as the surface of a sheet of paper. However, the top edge and the bottom edge have been identified^{2}. We’ll draw arrows on the sides to help us remember which edges have been identified.
Since the top and bottom are the same, we could take this piece of paper and roll it up to match up those edges. This leaves us with a tube of paper.
The two ends of the tube are the original left and right end of the screen, and so they are identified. Unfortunately, paper isn’t stretchy, but if you could bend it again, we could bring those two ends together to form the surface of a donut!
A mathematician would call the shape a torus.
When we are looking at the shape like this, we are thinking of a two-dimensional torus sitting inside of a three-dimensional space. But looking back at the game, there’s nothing about it that requires three dimensions. Even if the torus were sitting in three-dimensional space, the spaceship and the asteroids would have no way of realizing it.^{3} They can go up and down, left and right, but they have no way of interacting with the third dimension we visualized them in.
This turns out to be a key observation, and the basis for the mathematical idea of a manifold.
A manifold (with, say, 2 dimensions) is a shape where you can always, and only, go in two directions and the directions between those. You can go, say, up and down and left and right, as well as diagonally, but there is not even a conception of forward and backward through the two-dimensional screen. This is exactly what the game area for Asteroids looks like.
Another way to say this is that a 2-D manifold is a shape that locally looks like the standard 2-D space: a plane, like a piece of paper. So, in Asteroids, if we zoom in on an area just around where the ship is, the area around the ship also looks like a small piece of everyday two-dimensional space, the plane.
Yet another way to say this is that a manifold is a shape that locally can be described using as many coordinates as the manifold has dimensions. So, in Asteroids, the coordinates could be the coordinates on the screen. And a 2 by 2 matrix is a four-dimensional manifold, since there are four numbers.
These three different ideas of a manifold all mean the same thing, though it may take some thinking to see why. The technical definition, while important when actually working with these things, doesn’t really add much to our intuition, and so we won’t worry about it here.^{4}
The idea of purposely not thinking of these shapes inside of some high-dimensional space is surprisingly important and useful. (For example, thinking of our donut-torus as a two-dimensional surface rather than in three-dimensions or above.) But, before we get into that, let’s go through a few more examples of shapes that are (or are not) manifolds.
The most basic example is just n-dimensional space itself. You look closely at it and, unsurprisingly, it still looks like n-dimensional space. (n here is a number like 2 or 3 or 6.) Because this space is so important, I’ll finally give it it’s usual name, (read “R n.”) This is the prototypical manifold, and the model on which the rest are built.
Another example is the surface of a sphere. As on the surface of the Earth, there are two coordinates: latitude and longitude. We have two directions we can go: north/south and east/west. If you look closely, or, in other words, if the sphere is large enough, it looks like .
A 1-dimensional manifold has only one direction to travel, and so it has to (locally) look like a line. It turns out that the only examples of this are either a line or a circle^{5}.
Let’s make sure we understand the term circle. If you’re imagining this from your Sesame Street days:
Sorry, to break it to you, but a mathematician would not call this a circle. A mathematician would call that a disc… or a two-dimensional ball. You know, ‘cuz it’s filled in.
Now THIS is what a mathematician calls a circle. Just a line closed in on itself. NOT filled in. The space inside has nothing to do with the circle.
Of course, they might also call this a one-dimensional sphere…
Anyway.
What about an infinity figure? Is that a one-dimensional manifold?
If you focus at most points of this infinity, it looks like a line. So far so good.
However, if you focus on the point where the two parts cross, the shape instead looks like a cross, which is a problem if you were hoping to call it a manifold. At that point, there two directions you can go, but you can’t go in directions between those two directions. So it isn’t quite one-dimensional there, nor is it quite two-dimensional…
The infinity figure is not a manifold.
Most simple examples of shapes that aren’t quite manifolds are of this kind; most of the shape is like a manifold, but there are some places where these parts of manifolds come together is crosses. Another example would be two spheres just barely touching.
Now, back to the question: Why do we try so hard to think of manifolds as not sitting inside of some space like ?
There are two main reasons, as I see it. First, it lets us understand some objects that aren’t obviously “shapes” in a new way. For instance, the set of 2 by 2 matrices with determinant one^{6} (sometimes called ) turns out to be a three-dimensional manifold, not a four dimensional one, since the matrix can be written as and so only has three coordinates. If we can understand manifolds better, it will tell us something about matrices!
The second main reason for not wanting to think of manifolds in, say, three-dimensional space, is that how you draw your shape implies things about your manifold that are not intrinsic to the manifold.
For instance, look at a flat sheet of paper. If I were a two-dimensional being living on that paper, and you picked up the paper and bent it a bit, nothing that I could perceive from the two dimensional paper would let me know you bent it.
By thinking of the paper in three dimensional space, we can see the curving of the paper. And there are lots of cases where we might care about that, but we are also interested in what we can say about a shape independent of how we put it into .^{7}
Another piece of information we actually want to ignore that would be forced on us if we put our shapes in is distance (or length). As far as the definition of manifold is concerned, a circle (circle, not a disc!) or a horseshoe shape should be the same. And ignoring lengths, they are. (If you have trouble accepting that, just pretend these are made of yarn. You could easily rearrange the horseshoe to be a circle.)
It might seem that there isn’t much we can tell about a manifold without distances, but it turns out there is a lot that can be said. For instance, if you draw a circle (again, circle, not a disc!) on the surface of a sphere, there is a clearly defined inside and outside.
But if you draw the right circle on a torus, there is no inside or outside.
The study of shapes ignoring things like distances and angles is a whole field of study, called (manifold) topology.
But we like distances. So, in a coming post, we’ll talk about a way to define distances on manifolds, without thinking of the shapes in .
First, though, to satisfy my love of manifolds, we’ll have to have a post exhibiting a menagerie of weird and exotic manifolds.
- The odds of success are approximately three thousand seven hundred and twenty to one! ↩
- Identified is the technical term, meaning we treat a point on the top edge as the same point as the corresponding point on the bottom edge. ↩
- Unless, they could “look up,” if you will, and see into the other dimension. ↩
- The technical definition is that a manifold is a “topological space,” which means that you can say what “close” means, that is locally Euclidean (as per our second intuitive idea), second countable and (usually) Hausdorff. Second countable and Hausdorff are technical conditions that don’t come up that often. For instance, Hausdorff means that two separate points can’t be “infinitely close,” which only happens in kind of weird situations anyway. ↩
- The technical term for this is simple closed path. The simple means it doesn’t cross itself. ↩
- In case you aren’t familiar or have forgotten, a matrix is just a rectangle of numbers, in this case 2 by 2, so 4 total numbers. The determinant of a matrix is calculated from these numbers and tells you something important about the matrix (if it is invertible), though what the determinant is precisely doesn’t matter here. ↩
- The technical term for how we think of the shape inside of is the embedding of the shape. You can think of embedding as a way to draw a shape into space such that the shape doesn’t intersect itself. An immersion is drawing of the shape that lets it intersect itself. So, an immersion of a circle could be the infinity sign. ↩