Couple questions: What happens if you “attach” opposite sides of other solids, like an octahedron, dodecahedron, or icosahedron? I assume that, since an octahedron has the same symmetry as a cube, it would produce case 5 (a 3-torus). Would the octahedron and dodecahedron produce case 10?
What suggests that the universe is orientable? Is it something to do with kaon decay and CPT symmetry? If a particle did traverse a non-orientable universe, would it return merely as it’s mirror image (parity), or would charge and time reverse too?
Most solids, you can’t attach opposite faces in a consistent way. For one thing, the faces need to meet at an angle which is some integer fraction of 360[sup]o[/sup], or you’re going to end up with curvature singularities at the edges where you try to fit too many or two few pieces together, and you’re not supposed to be able to tell where the edges are. In fact, the cube is the only Platonic solid which will work. In every case where you can attach opposite faces, you’ll end up with one of those 18 basic cases, possibly in a hard to recognize form.
Unless you’re in a curved space. For instance, if you take a regular octagon, and glue opposite faces together, you’ll end up with a (two-dimensional) negatively-curved surface. You can get away with gluing octagon edges together, because in a negatively curved space, the angles of the octagon are less than 135[sup]o[/sup], and in fact, for one of the proper size, the angles would only be 45[sup]o[/sup] each (so you can fit eight angles around each vertex). There are also three-dimensional tilings which can be used for negatively-curved space, but unfortunately they’re all too complicated to describe in words (and damn hard even in pictures).
And the fact that P is not a complete symmetry suggests that the Universe is orientable, since a particle which traversed such a universe would presumably only reverse P (if it reversed C and P, then the fact that CP isn’t complete either would work just as well). It’s possible that beta decay occurs one way or the other in different regions of the Universe, in patches, but this would break the assumption of homogeneity.
Most solids, you can’t attach opposite faces in a consistent way. For one thing, the faces need to meet at an angle which is some integer fraction of 360[sup]o[/sup], or you’re going to end up with curvature singularities at the edges where you try to fit too many or two few pieces together, and you’re not supposed to be able to tell where the edges are. In fact, the cube is the only Platonic solid which will work. In every case where you can attach opposite faces, you’ll end up with one of those 18 basic cases, possibly in a hard to recognize form.
Unless you’re in a curved space. For instance, if you take a regular octagon, and glue opposite faces together, you’ll end up with a (two-dimensional) negatively-curved surface. You can get away with gluing octagon edges together, because in a negatively curved space, the angles of the octagon are less than 135[sup]o[/sup], and in fact, for one of the proper size, the angles would only be 45[sup]o[/sup] each (so you can fit eight angles around each vertex). There are also three-dimensional tilings which can be used for negatively-curved space, but unfortunately they’re all too complicated to describe in words (and damn hard even in pictures).
And the fact that P is not a complete symmetry suggests that the Universe is orientable, since a particle which traversed such a universe would presumably only reverse P (if it reversed C and P, then the fact that CP isn’t complete either would work just as well). It’s possible that beta decay occurs one way or the other in different regions of the Universe, in patches, but this would break the assumption of homogeneity.
I had only been considering that you would need a solid with an even number of sides. It didn’t occur to me that the angles have to add up – sort of like the reason there can only be five Platonic solids. The tiling of 3-space is a much more helpful way to look at it in this case than all the twisting and gluing. Thanks.
Last question: Where the hell do you learn this stuff?! What course do they cover this stuff in, or, more realistically, what book(s) could you recommend to someone who thinks this is really cool, is a practicing scientist, but doesn’t have an advanced math degree?
“inversion” - so we would have a universe where if you travel far enough you reach a mirror image world. It seems to me that this would give you, in effect, at least P and T inversion. If you passed a sign on your left early in your journey where you had the letters A first and then B second you would eventually pass the sign again but it is now on your right and you pass the B first and then the A. Doesn’t this amount to going back in time?
I’m not sure how that would be back in time… Care to elaborate further?
And in any event, a T inversion would be seriously wonked out (to use the technical term), because even aside from the subatomic one, the Universe has other arrows of time. Specifically, there’s the thermodynamic arrow of time (the future is the time when there’s more entropy) and the cosmological arrow (the future is when the Universe is bigger-- Obviously this one could potentially change, but it’s good for now, at least). It would be Really Weird, to say the least, if you made a trip around the Universe, and when you came back, broken eggs were unfalling up off the floor and re-assembling themselves. Even weirder, at some point in your journey you would have to pass a point where eggs stop breaking and start un-breaking, and that would be a pretty serious inhomogeneity.
As for where I learned this stuff, mostly from “water-cooler”-type conversations, and seminars presented by other folks working more actively in such fields. They had to pick it up from somewhere as well, presumably more formally, but I’m not sure where.
Ok, I was simply using the idea that as you travelled into the inverted region you would eventually pass the things you saw at the start of the journey but in reverse. I assumed this is what you meant by “inversion” at least. The idea that the basic cell is repeated in one dimension as a mirror image of itself. An object near the “edge” of the cell is soon re-encountered but inverted in the direction of your travel. But as I think about this I don’t like it since it seems to make some objects closer to the “edge” than others. So I think I probably haven’t understood your “inversion” idea. I get the idea of a 3-d analog of the mobius strip and that is what you mean by a “twist” in the way things are glued together. But how do we get inversions?