Understanding Dimensions in Cosmology

In discussing the large-scale structure of the cosmos,

astronomers sometimes say that space is curved

or

that the universe is, uh, finite but unbounded.

Whatever are they talking about?

Let's imagine that we are perfectly flat, I mean

absolutely flat, and that we live, appropriately

enough, in a flat land, a

land designed and named by Edwin Abbott, a

Shakespearean scholar who lived in Victorian England.

Everybody in Flatland is, of course, exceptionally flat.

We have squares, circles, triangles, and we all

scurry about, and we can go into our houses and do

our flat business.

Now,

we

have width and length, but no

height at all. Now, these little cutouts have some little

height,

but let's ignore that. Let's imagine that these are absolutely flat.

That being the case,

we know, us Flatlanders, about left-right, and we know about

forward-back, but we have never heard of

up-down. Let us imagine

that into Flatland, hovering above it,

comes a strange three-dimensional creature which,

oddly enough, looks like an apple.

And the three-dimensional creature sees an attractive,

congenial-looking square,

watches it enter its house, and decides, in a

gesture of interdimensional amity, to say

hello.

"Hello," says the three-dimensional creature. "How are you?

I am a visitor from the third dimension."

Well, the poor square

looks around his closed house, sees no one

there, and what's more, has witnessed a

greeting coming from his insides, a voice from within.

He surely is getting a little worried about his sanity.

The three-dimensional creature is unhappy about being considered

a psychological aberration, and so he

descends to actually enter Flatland.

Now, a three-dimensional creature exists in

Flatland only

partially. Only a plane, a cross-section through

him can be seen. So when the three-dimensional

creature first reaches Flatland, it's only the points of contact

which can be

seen, and we'll represent that by stamping the apple

in this ink pad and

placing that image in Flatland.

And

as the apple were to descend through, slither

by Flatland, we would progressively see higher

and higher slices,

which we can represent by cutting

the apple.

So the square,

as time goes on,

sees a set of objects mysteriously

appear from nowhere

and inside a closed room and change their shape

dramatically. His only conclusion could be

that he's gone

bonkers. Well, the apple might be a little annoyed at

this

conclusion, and so

not such a friendly gesture from dimension to dimension,

makes a contact with the square from below and sends our

flat creature fluttering and spinning above Flatland.

At first, the square has no idea what's happening.

He's terribly confused.

This is utterly outside his experience.

But after a while,

he comes to realize that he is seeing inside

closed rooms in Flatland. He is looking

inside his fellow flat creatures.

He is seeing Flatland from a perspective no one has ever seen it before, to

his knowledge. Getting into another dimension provides,

as an

incidental benefit, a kind of X-ray vision.

Now our flat creature slowly descends to the surface,

and his friends rush up to see him.

From their point of view, he has mysteriously appeared from nowhere.

He hasn't walked from somewhere else.

He's come from some other place. They say, "For heaven's

sake, what's happened to you?"

And the poor square has to say, "Well, I was in some

other mystic dimension called Up."

And they will pat him on his side and

comfort him, or else they'll ask, "Well, show us.

Where is that three-dimen- third dimension?

Point to it." And the poor square will be unable to comply.

But maybe more interesting is the other direction in

dimensionality. What about the fourth dimension?

Now, to approach that, let's consider a cube.

We can imagine a cube in the following way.

You take a line segment and move it at right angles to itself an

equal length. That makes a square.

Move that square an equal length at right angles to itself, and you have a

cube.

Now,

this cube, we understand,

um, casts a shadow.

And

that shadow we recognize. It's, you know,

ordinarily drawn in, uh, third-grade classrooms as two

squares with their vertices connected.

Now, if we look at the shadow of a

three-dimensional object in two dimensions, we

see that in this case, not all the lines appear

equal. Not all the angles are right angles.

The three-dimensional object has not been perfectly represented in its projection

in two dimensions, but that's part of the cost of

losing a dimension in the projection.

Now,

let's take this three-dimensional cube

and project it, carry it through a fourth physical

dimension. Not that way, not that way, not that

way, but at right angles to those three

directions.

I can't show you what direction that is, but imagine that there is a fourth

physical dimension.

In that case, we would generate a four-dimensional

hypercube,

which is also called a tesseract.

I cannot show you a tesseract because I and you are

trapped in three dimensions. But what I can show you is the

shadow in three dimensions of a four-dimensional

hypercube or tesseract. This is it.

And you can see it's two nested cubes, all the vertices

connected by lines. And now the

real tesseract in four dimensions would have all the

lines of equal length and all the angles right angles.

That's not what we see here, but that's the penalty of

projection.

So you see, while we cannot imagine the world of

four dimensions, we can certainly think about it

perfectly well.

Now,

imagine a universe just like Flatland, truly two-dimensional

and entirely flat in every direction,

but with one exception.

Unbeknownst to the inhabitants, their two-dimensional universe is

curved into a third physical dimension,

maybe into a sphere, but at any rate, into something entirely

outside their experience.

Locally, their universe still looks flat enough,

but if one of

them, much smaller and flatter than me, takes

a very long walk along what seems to be a straight line,

he would uncover a great mystery.

Suppose he marked his starting point here

and set off to explore his universe.

He never turns around and he never reaches an edge.

He doesn't know that his apparently flat universe is

actually curved into an enormous sphere.

He doesn't sense that he's walking around a globe.

Why should his space be curved? Because there's so much matter in this

universe that it gravitationally warps space, closing it back on

itself into a sphere. But our Flatlander doesn't know this.

After a long while, he'll find he somehow returns to his

starting

point.

There must be a third dimension. Our

Flatlander couldn't imagine a third dimension, but he

could sure deduce it. Now, increase all the dimensions in this story

by one and you have something like the situation which many

cosmologists think may actually apply to us.

We are three-dimensional creatures trapped in three

dimensions. We imagine our universe to be flat in

three

dimensions, but maybe it's curved into a

fourth.

We can talk about a fourth physical dimension, but we can't experience it.

No one can point to the fourth dimension.

I mean, there's left, right, and there's forward, back, there's up,

down, and, uh,

there's, uh, some other direction simultaneously at

right angles to

those familiar three dimensions.