Well Bill S. I'm not sure just how to reply. The problem to me is that what intuitive feel I have for 1+1D and 2+1D universes is very contaminated by my automatic understanding of a 3+1D universe, since that is what I am living in. But I will try to interpret what I have read about String Theory, and its requirements for more dimensions.
They say that the extra dimensions are there, but that they are "compactified", so that they are too small to be detected. The normal way they present it is that the compact dimensions are rolled up so that their extent is very small. Actually it may not be 'rolled" which implies something like a roll of cloth, but that is the way it is usually explained. They have some ideas about the way it is actually folded, but saying rolled is a shorthand way of describing it in a non-mathematical way. Now a rolled up dimension is kind of hard to visualize. I have had a hard time myself figuring out what it means to have it rolled up. I look at what I mean when I say something about the dimensions of something. I mean that I can travel along one dimension, for example North, then make a turn and travel off in a different dimension maybe East, or along a different dimension which would be up or down. So basically a dimension is a direction in space. Now they tell me that there is another dimension at right angles to all three of the dimensions I have been traveling, but I can't see or travel in that dimension because it is too wrapped up in itself. This is not intuitive, we just have to accept it because we are told it is so.
Now let me see if I can figure out what it would mean to "get from zero volume to a positive volume". Let's just forget the time dimension for this discussion. Well, for one dimension the energy of the universe would be concentrated in a straight line with no thickness. That is a lot easier to say than it is to visualize. But the only interactions would be between adjacent blobs of energy. I'm not sure how differentiated the energy would be. If we assume that an uncertainty principle existed, and I think it probably would, then the energy would exist in very small blobs. Presumably at the Planck scale. I can visualize this as knots in a string, but of course my visualization is a 3 dimensional visualization of a 1 dimensional universe. Anyway the energy density of the universe would determine the size and spacing of the blobs. If it was high they would be large and close together, as the universe expanded they would be able to spread out more. I envision the addition of the second dimension as the unfurling of a sail from a mast with a roller boom. That is a boom that has the sail wrapped around it. When you want to furl the sail you rotate the boom so the sail wraps around it. To unfurl it you just turn the boom the other way and the sail spools off. So for 1 dimension the energy would be running back and forth along the boom. As the second dimension unfurled the energy would get to jump off of the boom and spread itself all over the sail. So now instead of just running back and forth the energy could spread out over the second dimension. So now we have to talk about the energy density of an area, rather than a line. In going from 2 to 3 dimensions the sail analogy breaks down, because we have to picture the 3rd dimension rolling out from the first 2 dimensions. I think maybe the transition to the 3rd dimension can be envisioned as a ballon. Before you blow up a baloon you can lay it out flat and it will look kind of like a 2 dimensional surface. Of course the 3rd dimension is there, it is just kind of rolled up or "compactified" so that there is no access from the 2 dimensional surface. But as soon as it starts to inflate then you can suddenly discern the 3rd dimension and the energy that was trapped in the 2 dimensional universe can spread out into 3 dimensions.
Ok, let's see if that helps at all. And keep in mind that this is only my mostly uninformed raving. So if that isn't how it is don't blame me.
Bill Gill
