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<   No. 1280   2006-07-29   >

1 {scene: Adam and Death of Inhaling Hatmaking Chemicals are walking across the Infinite Featureless Plane of Death.}
1 Adam: This Infinite Plane of Death With One Feature...?
1 Death of Inhaling Hatmaking Chemicals: YES, GUV?
1 Adam: What's its geometry?
2 Death of Inhaling Hatmaking Chemicals: A FFREE-DIMENSIONAL RIEMANNIAN MANIFOLD WIFF A SLIGHT NON-EUCLIDEAN CURVATURE.
3 Adam: Positive or negative?
3 Death of Inhaling Hatmaking Chemicals: POSITIVE, I FFINK...
4 Adam: Aha! So it's not infinite either!

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Hmmm... let me think about this one for a while...

Okay. Let's try this:

A manifold is basically a fancy formal mathematical treatment of what we would call "space" (in the sense of the volume we occupy and move around in, not outer space)*. A Riemannian manifold is just a manifold with a bunch of properties that make it nice in a formal mathematical sense, specifically which allow you to measure things like length and volume in a sensible way. (A Riemannian manifold is actually a simpler object to understand than a general manifold, because it behaves the way we intuitively think space should behave.)

Now, a manifold can either be Euclidean or Non-Euclidean. These fancy words refer to the ancient Greek mathematician and geometer Euclid, who formulated pretty much all those rules of geometry you learnt and promptly forgot at school. The most important one for this discussion is the idea that parallel lines never meet or intersect. Pretty basic and obvious stuff, since parallel lines are always the same distance apart. If you're using Euclidean geometry, that is...

Non-Euclidean geometry involves straight lines that stay the same distance apart if you look at them here, but if you look at them a few miles, or light years, down the road, they actually get closer together (or further apart). Not because they're drawn inaccurately or curved, but because the space (or the manifold) in which they exist is itself curved.

Think of it this way: If you have a pair of parallel lines that never intersect, and you want to make them intersect a few miles away, you can either curve the lines, or leave the lines straight and curve the space they pass through. Yes, it's basically as simple as that.

Now, if you're going to all this bother of curving space, there are two different directions you can curve it. You can curve it so the parallel lines intersect; this is called positive (or elliptical) curvature. You can also curve it the other way, so the parallel lines get further apart; this is called negative (or hyperbolic) curvature.

Still with me? We're nearly there. The interesting thing about space is that it goes forever. Ah... Well, it does if it's Euclidean (i.e. those parallel lines stay the same distance apart, remember). It also does if it's hyperbolic. But if it's elliptical, something odd happens. If you follow a straight line forever, eventually you "wrap around" and come back to where you started from, from the other direction. Seriously. Just like a video game.

What this means is, if you live in a manifold which has a positive non-Euclidean curvature, space is not infinite. Because no matter how far you travel, you never get more than a certain distance away from your starting position before you start getting closer again and eventually return.

The mind-boggling thing is that all this isn't just mathematical gibberish. The space we actually live in is a Riemannian manifold. And it may well be curved. We're not sure if it is curved yet (and if so, positively or negatively), because if it is, then the curvature is so tiny that it's very difficult to measure. We can only measure it by looking at objects a really long way away. This is what some astronomers do... They make observations to try to measure the curvature of our universe.

If you ever wondered just what the heck astronomers are trying to achieve with their research funding - now you know.

* Manifolds don't have to be three-dimensional like the space we're familiar with, but that's not important here.

John Armstrong of the Mathematics Department at Yale University writes:
Evidently Jamie forgot that you need delta-pinched positive curvature to conclude that a surface is diffeomorphic to S2. That is there are constants c and δ so that at every point the curvature k satisfies
0 < δc < k < c
Consider the paraboloid z = a(x2+y2), along with the obvious map from the x-y plane up to it. Define a metric on the x-y plane by pulling back the metric induced on the paraboloid by the standard Euclidean metric in R3. This has everywhere-positive curvature bounded above by 4a2, but it gets arbitrarily small as you move out.

Actually, that might make the origin a good place to put the desk...

Now, while I understood this, I wasn't quite sure how to explain it in simpler terms. Then I got another e-mail from Daniel Bartlett of the University of Arizona:
It is possible to have a manifold with positive curvature everywhere, complete, and not compact, thus infinite. The only detail is that the curvature cannot have a positive lower bound - as you go out far enough, it approaches 0; if the curvature is bounded below by some positive constant, the manifold (and its universal cover) must be compact.

Of course, the "Infinite Featureless Plane of Death" looks fairly homogeneous, implying the same curvature everywhere, but it could simply be a very slight nonzero curvature, and it could vary, imperceptibly by human standards. And why should Deaths know differential geometry?

This is a simpler explanation of the same thing. To put it as simply as I can express it, if the space is positively curved, but gets less curved as you move away from some central location, so that it gets closer to Euclidean but never quite actually reaches it, then it's possible to have the space be positively curved everywhere (by different amounts, depending on where you are) and still be infinite. The drawback of this is that there must be some central point where the curvature is a maximum, so the space wouldn't be uniform - and maybe this is where the Head Death's desk is.

To picture what the "plane" would look like in this case, you can imagine a giant paraboloid of revolution - like a parabolic reflector dish that continues forever. The tip of the paraboloid is the point of maximum curvature, where the Head Death's desk might be.

2016-01-13 Rerun commentary: Here's another way to think about it. Think about a manifold in just two dimensions, rather than three.

A non-curved Riemannian manifold in two dimensions is just a flat surface, like a tabletop, only extending indefinitely, forever in all directions. We know what a tabletop is like, and it's not too hard to imagine it going on forever in all directions.

If you built a model railway on such a tabletop, you would lay your parallel railway tracks and you could extend one line of track forever into the distance. The tracks, being parallel, would stay the same distance apart and never meet, no matter how far you travel along them. The rails which make up the track are all three of straight, parallel, and equidistant

But there's another type of two-dimensional manifold that is also fairly easy to think about - the surface of the Earth. If we ignore the local topography like mountains and valleys and oceans and stuff, we can basically think of the Earth as a sphere. The surface of this sphere is a two-dimensional manifold with a uniform positive curvature. But the curvature is quite small compared to the scales we encounter around us.

Imaging building a model railway on the surface of this idealised spherical Earth. Inside your house, this is essentially the same as building it on a flat tabletop. The curvature of the Earth is so tiny as to be imperceptible.

But, if you build a long straight track and extend it into the distance... eventually it will wrap all the way around the world and return to your house from the other side! This is exactly the property I was talking about above - of wrapping around like a video game! Head in any direction from your home, and just keep walking, and eventually you'll find yourself approaching your home again from the other side. Just like Pac-Man!

But what about those train tracks? We know the tracks are the same distance apart, because the train wheels have to fit in them. They are equidistant. But didn't I say that straight lines in a positively curved space which may look locally equidistant and parallel in one location eventually meet? I did! Well, here's the rub: In a positively curved space, the rails of a straight piece of train track are not straight lines.

If you draw actual straight lines and extend them along the surface of the Earth[1], they will intersect. In fact, they will intersect once you travel a quarter of the way around the world. And then intersect again three-quarters of the way around.

An easy way to see this is to imagine you are at the equator of the Earth, and you draw straight lines pointing north and south. The lines might be a few centimetres apart, like model railway track rails, or a few hundred kilometres apart. The thing about being at the equator and drawing lines that point north and south is that it's easy to point the lines, You just aim them north and south, using whatever means you have of determining north and south accurately (navigation charts, the stars, GPS, whatever).

Okay, so we have a pair of straight lines at the equator, both of the lines pointing north/south. Sounds like they're parallel, right? Now let's follow them. Let's go north. The thing about lines on the surface of the Earth that point north is that they point towards the north pole, right? So we travel a quarter of the way around the Earth and we end up at the north pole. Here's the thing: BOTH of the straight lines that we drew pass through the north pole. They have to - they both point north. They look parallel when we started them at the equator, but now they pass through the same spot, in other words they intersect.

Straight lines on the surface of the Earth that look parallel at one point eventually intersect somewhere else.

If we keep following the lines, they'll diverge again after we pass the north pole, and on the other side of the Earth from where we started, when the lines cross the equator again, they are now as far apart as they were when we started - a few centimetres or kilometres or whatever it was. Keep going and at the three-quarter way mark we are at the south pole, where the lines intersect once again. And them eventually we return where we began, and each line meets up with itself from the other side, like Pac-Man.

Straight lines drawn on the surface of the Earth are actually, if we think in three dimensions, giant circles that go around the world. We call them great circles.

What about "straight" railway track rails? We think of them as parallel and straight, but as we have just seen, truly straight lines on the surface of the Earth intersect if we extend them far enough. Railway rails are not allowed to intersect - they have to stay the same distance apart to allow the trains to run on them. The conclusion is that railway tracks on the surface of the Earth are not actually straight. They are ever so slightly curved, even the dead "straight" ones, so that they don't intersect like straight lines would.

Another simple way to realise this is to start a track at the equator, pointing north. The rails locally look straight. But if they were straight and point north, then both rails would pass exactly over the north pole. But to be a railway track, the rails can't intersect at the north pole like this, so they must be curving apart ever so slightly the whole way, to maintain equidistance.

Cool!

[1] I mean "straight" in the sense that the lines don't swerve left or right on the surface of the Earth. The lines curve around the surface of the Earth, but in this example that is defined to be straight. "Straight" on the surface of the Earth means "walk in a straight line", staying on the surface, not floating off into space.

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