(4) Tue Jun 20 2006 23:16 Extra Spatial Dimensions: I have a question about these little buggers. Kris couldn't answer it, so you know it's tough. Whenever I heard an explanation of string theory or any other theory that predicts more spatial dimensions, they always said the extra dimensions were "rolled up very small". That doesn't make sense. Rolled up very small through what dimensions?

Then I think I figured it out with a Flatland analogy. Suppose Flatland exists in a 3D universe, but one in which nothing is more than a Planck distance tall. The Flatlanders extend in three dimensions, but they only perceive two. They would say that the third dimension was "rolled up very small", even though that doesn't make sense.

Wikipedia seems to back me up on this one, ladies, so the first question is, am I right? Does the analogy hold? Second question is, what is with the "rolled up" nonsense? Are they trying to convey that the structure of space makes it impossible for something to be large in that dimension ("the universe measured along these additional dimensions is subatomic in size")?

- Comments:
Posted by Ian Bicking at Wed Jun 21 2006 01:31

If it wasn't rolled up, and it was small, then wouldn't it have boundaries? I think that's what the metaphor is for. Like the flatworld put onto a sphere or globe, it is finite but has no start or end.

Or maybe it would be like a 2 dimensional world on a very long, skinny cylinder. It would seem 1 dimensional, because everything would nearly always collide on the when it went up and down the cylinder. But there would actually be this other very constrained dimension. It wouldn't have a start or end or height, but it would be rolled up.

So there is an extra dimension, in that we find that extra dimension convenient to create metaphors about the topology.

At least, that's the metaphor I'm pulling out of my ass tonight.

Posted by Tim May at Wed Jun 21 2006 06:32

The "long, skinny cylinder" analogy is the one I remember from, oh, some popular work on the subject, probably Michio Kaku's

*Hyperspace*. Certainly it's the only way "rolled up" has made any sense to me.Posted by AndrĂ© Roberge at Wed Jun 21 2006 07:09

To answer your question properly would take more than a few pages of text... I'll just limit myself to some pointers.

"Rolled up very small through what dimensions?" you ask. What you are picturing (and the image authors often want to convey to you) is the extrinsic curvature of an embedded object i.e. the image that comes to your mind is that of a curved object (e.g. rolled up cylinder) as viewed from outside. However, in string theory, there is no outside dimension from which you can view the object.

Think of designing an asteroid game on your computer (assuming you are old enough to know about this game). When an asteroid leaves the right edge of the screen, it reappears on the left edge. It is as though the screen has no boundary. The same is thought to be true of these extra dimensions. The problem is (from our crude senses) that the extent of the "screen" is so small that we can't perceive it - we certainly can not move through it as we do in the usual 3 dimensional spaces. However, we "feel" these extra dimensions in some other way. For example, in an old theory (Kaluza-Klein) that has *a few* similarities with string theory, an attempt was made to describe the world as having 4 spatial dimensions and 1 time dimension, with one spatial dimension being periodic ("rolled up") and "small". So, the position of an object at time "t" is given by (x, y, z, u). If you do the math, you find out that the effect of the extra dimension looks exactly like what we know of electro-magnetism i.e. it's mathematically equivalent to the usual description of electro-magnetism in our 3 spatial dimensional world.

Is our world 3 dimensional (spatial) with electric and magnetic field present... or is it 4 dimensional (spatial) with one spatial dimension rolled up but no explicit magnetic and electric field added?... If they are equivalent mathematically, how do you decide?

Posted by uncle pedro at Wed Jun 21 2006 12:26

wait a minnit... are we talking about physics or is this a frank chu lecture?