Manifold Menagerie

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.

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.

torusfp

torus

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.

kleinfp

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!

asteroidsklein

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.

foldingaklein

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 \mathbb{R}^2!

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, \mathbb{R}^3!

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:

string

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, \mathbb{R}^2.

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 \mathbb{R}^3. 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?

mindblown

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 n dimensions, this visualization may require as high as 2n dimensions. Our Klein bottle (with n=2) is an example of this, as it requires 4 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.

kleinwalk

Trippy.

Okay, the Klein bottle is weird.5

Let’s talk about some more pedestrian6 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, \mathbb{S}^2 (read “S 2”), since there are two directions you can go.

Along with that, you have the one-dimensional sphere, \mathbb{S}^1. This has one direction you can go… in other words, it’s a circle!

We can go higher in dimensions as well. What is \mathbb{S}^3, 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 \mathbb{S}^2, horizontal slices of it are just circles, \mathbb{S}^1.

circles

So, as we take slices of \mathbb{S}^2, we have small circles that get larger till we pass the equator, then get smaller again till they disappear.

We can visualize \mathbb{S}^3 the same way. The problem is, it’s naturally in \mathbb{R}^4 instead of \mathbb{R}^3, so it’s harder to see. To help us, let’s think of \mathbb{R}^4 as the three normal dimensions, plus time.

spheres

So, if we take \mathbb{S}^3, 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, \mathbb{S}^2, 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 \mathbb{T}^2 (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, \mathbb{T}^3.

cubes

Pretty cool. Of course, it’s much harder to visualize as sitting inside of \mathbb{R}^4. 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.

2torusfp

Now, it’s much harder to see what’s going on without a hint. What is this shape?

2torus

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.

3torus

3tor

Let’s see, what else weird can we do?

hmmm

Oh, I know!

Okay. Take the sphere (\mathbb{S}^2). Now, for each point, identify it with the point on the exact opposite end of the sphere.

rp2visualization

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!

rp2hemisphere

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 \mathbb{RP}^2 (read “R P 2”), where the \mathbb{RP} 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:

spherepathpinching

However, if we took the equivalent path on \mathbb{RP}^2, 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.

rp2singlepath

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.

doublepathrp2

So, in \mathbb{RP}^2, one closed path is stuck. But if you wrap it around twice, all of a sudden you can shrink it down to nothing!7

mindblown

So this new space is not a sphere!

Again, we call this space \mathbb{RP}^2. 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!

fp

Like the Klein bottle, \mathbb{RP}^2 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.

crosscap
By Maksim, Wikimedia Commons

\mathbb{RP}^2 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 \mathbb{R}^3.


  1. 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. 
  2. Remember, a circle is the outer edge, not the filled-in shape. 
  3. A visualization of a manifold (in \mathbb{R}^n, 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. 
  4. In fact, a Klein bottle is two Möbius strips with their edges glued together. 
  5. 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! 
  6. Pun! Ba-dum-ching! 
  7. 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… 
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Asteroids on a Donut

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.

homersimpson

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 identified2. We’ll draw arrows on the sides to help us remember which edges have been identified.

flat

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.

tube

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!

torus

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.

zoomin

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 (x,y) 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, \mathbb{R}^n (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 \mathbb{R}^2.

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 circle5.

lineandcircle

Let’s make sure we understand the term circle. If you’re imagining this from your Sesame Street days:

toddlercircle

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.

circle

Of course, they might also call this a one-dimensional sphere…

Anyway.

What about an infinity figure? Is that a one-dimensional manifold?

infinity

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.

twospherestouching

Now, back to the question: Why do we try so hard to think of manifolds as not sitting inside of some space like \mathbb{R}^n?

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 one6 (sometimes called SL(2,\mathbb{R})) turns out to be a three-dimensional manifold, not a four dimensional one, since the matrix \begin{bmatrix}a&b\\c&d\end{bmatrix} can be written as \begin{bmatrix}a&b\\c&\frac{1+bc}{a}\end{bmatrix} 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.

bent

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 \mathbb{R}^n.7

Another piece of information we actually want to ignore that would be forced on us if we put our shapes in \mathbb{R}^n 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.)

circleandhorseshoe

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.

circleonasphere

But if you draw the right circle on a torus, there is no inside or outside.

circleonatorus

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 \mathbb{R}^n.

First, though, to satisfy my love of manifolds, we’ll have to have a post exhibiting a menagerie of weird and exotic manifolds.


  1. The odds of success are approximately three thousand seven hundred and twenty to one! 
  2. 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. 
  3. Unless, they could “look up,” if you will, and see into the other dimension. 
  4. 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. 
  5. The technical term for this is simple closed path. The simple means it doesn’t cross itself. 
  6. 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. 
  7. The technical term for how we think of the shape inside of \mathbb{R}^n 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. 

No one liked Cantor

Over the last month, we spent a lot of time talking about infinity. Pretty much everything we discussed was the work of one man, Georg Cantor.

cantorgeorg

You might think that such a gifted mathematician, who laid the basis for all our modern understanding of infinity would have been universally respected by his contemporaries.

Not so!

Cantor’s ideas of infinity, as well as his methods of proof, infuriated many prominent mathematicians.

angrykronecker

Cantor was born in 1845 in St. Petersburg, though his family soon moved to Germany, seeking milder winters. His exceptional mathematical skills were noted early, and he studied with some of the famous mathematicians of the day, such as Kronecker and Weierstrass. (If you’re not a mathematician, you’ve probably never heard those names before, but, I promise, they’re well-known mathematicians!)

After graduation, he took up a professorship at the University of Halle. This was a good position, and he advanced to full professor at a young age.

Cantor’s early work was on number theory, but his famous work was on the nature of infinity, and the sizes of sets. In an early paper he proved that the set of real numbers is larger than the set of counting numbers (see this post). This was the first instance of showing that infinite sets could have different sizes.

Before Cantor, infinity was a vague thing, and often rejected as a proper thing. For instance, one of the most famous mathematicians of all time, Gauss, had the view that, “Infinity is nothing more than a figure of speech which helps us talk about limits. The notion of a completed infinity doesn’t belong in mathematics.”1

Indeed, many of Cantor’s contemporaries hated Cantor’s ideas. One of the most prominent opponents to his ideas was Leopold Kronecker, head of the math department at the University of Berlin. Berlin was more prestigious than Halle, and Cantor wanted to get a position there, but Kronecker’s opposition nixed any opportunity for that.

Kronecker’s position is often put under the label “finitism.” He once said, “God created the natural [counting] numbers; all else is the work of man.” The viewpoint is that the counting numbers clearly exist, but any other mathematical object should be able to be derived by a finite number of steps from this basic foundation. Kronecker rejected Cantor’s original proof that the set of real numbers is bigger than the set of counting numbers on this ground. Cantor developed the proof we discussed in response to his criticisms.

Kronecker at one point said, “I don’t know what predominates in Cantor’s theory – philosophy or theology, but I am sure that there is no mathematics there.” Among other things, he called Cantor a “scientific charlatan,” a “renegade” and a “corrupter of youth.”

corruptingyouth

Kronecker was not so fond of Cantor.

These criticisms, among others2, stung Cantor deeply, leading to several episodes of depression. It sucked the joy of math out of him for several years. For a while, he turned to, of all things, the Shakespeare authorship question, on which he wrote a couple of pamphlets.

He did eventually return to math, though his later work was not as transformative as his earlier work. Cantor eventually retired in 1913, and lived in poverty due to World War I. He died in 1918 at the age of 72.

Many mathematicians realized the importance of Cantor’s work. In 1904, Cantor was awarded the Sylvester medal, the highest honor of the Royal Society. Cantor’s research into infinity have influenced mathematicians for generations. The influential and important mathematician Hilbert said of his work, “No one will drive us from the paradise which Cantor created for us.”3

paradise


  1. Dunham, William. Journey through Genius: The Great Theorems of Mathematics. Penguin. p. 254. Gauss lived 1777 to 1855, dying when Cantor was 10 years old. 
  2. At one point, Cantor tried to publish a paper in Acta Mathematica, which was run by an acquaintance, Mittag-Leffler. Mittag-Leffler asked him to withdraw the paper, as the paper was “one hundred years too soon.” That… hurt their relationship. 
  3. Though, apparently, another mathematician replied, “If one person can see it as a paradise of mathematicians, why should not another see it as a joke?”