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 and circles and 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 a, 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 in

equal

length. That makes a square. Move that square in 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 a

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 u-