Ok, so here's what I think a 2D world would be like. Everything would be on a plain that has no depth, only width and length (I like to imagine it like this; If you were to somehow break free into a third dimension you would be looking at it like a picture, the view point from above). But, I can't figure out some things. Like; Objects could only move side to side, and foward and back (As up and down wouldn't exist). But because they have no depth, when two objects intersect, there would be nothing to stop both of them existing in the same point of space at the same time. My dad thinks that they would still be repelled from each other (Maybe by like the electrical charge, positive and negative, of the matter). Before I thought that an infinite amount of objects would be able to exist on the same point, however, now I think that the matter from the part of the object that was "covered" would cease to exist (As it couldn't be covered - there are only two dimensions, nothing can be "Behind" something else). ??? As well as this, My dad thinks that any life in this 2D world (if it could exist, which I don't think it could) would just see a line, but as every object would have no width, I think that from a viewpoint within the world, you wouldn't be able to see anything and any life would have to have some other way to sense. Anyone have thoughts on this??
There are other problems with a 2D world. Such as an inability for there to be gravity in a vacuum (at least according to General Relativity).
It works as it does in our universe, 'normal matter' can't be at the same point at the same time. There's no repelling force (other than the Pauli one) due to this but its simple to see. In 3 dimensions suppose you have an object X at (x1,x2,x3) and an object Y at (y1,y2,y3). They are at the same point if and only if x1=y1, x2=y2 and x3=y3. Reducing this to the 2 dimensional world you consider is simply a matter of making one of these three true. For instance, without loss of generality, you can say y3=x3=0. The regions described by (x1,x2,0) and (y1,y2,0) are both 2 dimensional and are coincident. The 3 dimensional conditions now reduce to the two dimensional ones, you need x1=y1 and x2=y2. Actual forces can exist but for EM and Newtonian gravity they go like \(\frac{1}{r}\), not \(\frac{1}{r^{2}}\). In fact this generalises for D dimensions to \(\frac{1}{r^{D-1}}\). In 3 spacial dimensions its the inverse square law. In 1 dimension it means the force is constant but this makes sense as the force cannot spread out, it just moves through the space. Any life would have to 'vomit up' its food since if it had a digestive tract all the way through it it'd be cut in half! Our bowels are a fundamentally 3 (at least) dimensional construct! You're right, our eyes see a 2d picture (ie I can formalise this if you want, have a gander at projective spaces) and we interpolate to 3d by the use of two eyes positioned slightly differently. In a 2 dimensional world they'd see only in lines and 2d would be inferred from having two eyes differently placed. How you'd get eyes and blood supplies in such a world I don't know but the principle is right. I suggest you watch this video. Its the irreplaceable Carl Sagan doing what he did so very very well, explaining the mysteries of the universe to an entire generation.....
