When thinking about the universe, there are some concepts that are hard to wrap your head around. This is particularly true when talking about the shape of the universe. For example, we sometimes talk about how the universe can be *flat, *but that it could* *also be finite in size *and* has no edge. How is that possible?

Let’s start off with a flat sheet of paper. This is your standard two-dimensional picture of the universe. The thing about our sheet of paper is that if we draw a vertical line on our paper, move to the left or right and draw another vertical line, our two lines are parallel. They will never cross no matter how big our sheet of paper is. This is one way to test for flatness. As a counter example, the Earth looks flat, but if we drew a north-south line, moved east and drew another one, our two lines would cross at the north and south poles. This means our roughly spherical Earth is not flat, even though it looks that way on the human scale. So now we have a geometrical rule: on flat surfaces, parallel lines don’t cross (they also don’t spread apart).

So our sheet of paper represents flat space. But, you might say, the paper has edges, does that mean there is an “edge” to space? The answer turns out to be no, and there are two ways we can get rid of our edge. The first is simply to imagine that our sheet of paper goes on forever. No matter how far you trace a line you will never reach an edge, because the paper keeps going on forever. This would be a visualization of an open or infinite universe.

The other way we can get rid of the paper edges is to “connect” each edge to another edge. You might imagine wrapping and taping the sheet of paper, but we don’t have to do that. All we need to do is make a rule: When you get to an edge of the paper, move to the opposite edge and keep going. So if we start to trace a vertical line, we will eventually get to the top of the page. There we follow our rule, move the bottom of the page and keep tracing up. Eventually we will arrive where we started.

Using our edge-connection rule, our paper is still “flat”. If we draw a vertical line (all the way around), then move to the right and draw another one, the two lines don’t cross. Of course this time our lines don’t go on forever, they just go round and round and round. This represents a universe that is both flat and closed.

If you imagine the sheet as rubbery, and really connected the sides following our rule, you would get a torus. A torus is “flat”, in that parallel lines don’t cross, but it doesn’t look flat until you unfold it.

So there you have it. We can use our edge rule to imagine a space which is both flat and closed. Just keep in mind, in space there is no real edge that warps you to the other side of the universe. We have to use our trick because we’re playing with paper, which is made of physical atoms and such. Space and time don’t have that limitation, so they can be connected in all sorts of ways and still be flat.

As for what that means cosmologically, I’ll leave that for another time.

## Comments

If the Universe has the topology of torus, even if its geometry is flat, it has some weird properties, which come directly from the fact that its homotopy group is nontrivial. Namely, you can have a loop of wire, that goes around the universe and you cannot tighten it. At least for me, this is the weirdest aspect of finite flat Universe with no edges… not that we have enough wire, of course.

I have seen the new mapping of the big bang, its surprising at a point. It appears to have three out ward flows in different directions. Multi Curvatures from one point. Where space and time and matter separate in other multi Universes. Its only passive gravitational forces bending photons on their billions of light years passing by stars, black holes ,quasars and galaxy’s it only appears to loop. When all the data from new instruments comes, a new reality will emerge. The outward flows shows spherical processes in astronomical quantum physicists believe in multi dimension. More Questions then answers. The 72 Quasar is a key to creating everything in the early Universe. This prediction of other dimensions (Known as equivlance classes) the determinant is y. The inverse divided by y this general theorem from a group theory every rotation has exactly 2 square roots. how ever their is one exception identity rotation has infinitely square roots 180 square root of 1. Making this short, mathematics, differential algebra product rule Calculus derivative two or more functions,flows in the action formation. Theoretical physics an analysis of flow a study (Gauge) (Symmetries) flows the formatation of a theory is invariant under! When multiplication two involutions 1 and -1 x squared 1 has At most 2 roots. Double angle formulas. Its attachments like infinite and finite need to mathematical work out.

There is, in four dimensions, a most unusual beast called a ‘tiger’. Its surface is topologically equal to a 3d prism, where the left and right, up and down, and backwards-forwards are joined together. Yet two of these circles are in perfect cartesian product.

In terms of ‘Poincare’s Dodecahedron’, i showed John Conway how to make it into five tetrahedra.

I wonder why this question is always answered the same way…there is no edge. Sorry but there is no way to known that. I think many want, for bias reasons, the existence of an infinite so they avoid a beginning or allow for multiverse, but just as they deny freewill, against all logic, they deny that infinity is just a concept that cannot be in reality.

There could be an actual edge. In fact, just because the UV is big you cant getaway from something obvious. When the UV was only as big as a football field….are you saying it didnt have an edge? Because that’s ludicrous. That means if you were there you would see this end of the universe.

So can we stop this nonsense and avoidance and address the question with more effort than that pat answer…like we couldn’t possibly comprehend anything.

Mathemticians, i found have difficulty with the distinction between unbounded and infinite and indefinite.

In GR there is no unique way to say wether space is finite or infinite, it can be finite for some observer, while infinite for another observer, All we know is that the universe is very very big.

You need 3 D space to rap the paper , the analogy brakes down . You might as well say we have n Dimensions , but they’re unobserved because we’re fixed to the surface . Things are even more complex with spacetime .

Actually you don’t. For a physical sheet of paper made of atoms and molecules, yes, but not spacetime itself.