A question about the flatness of the universe.

[$id]

A question about the "flatness" of the universe.

I may be out of my depth here or overthinking it but...

Does the fact that the universe is flat (angles in triangle add upto 180deg) mean that the complete universe has to have a edge?

If it had positive curvature, then it doesn't necessarily have to have a edge (like the surface of the earth)

Its just been bugging me that's all.

sid

 

[George Jones]

Does the fact that the universe is flat (angles in triangle add upto 180deg) mean that the complete universe has to have a edge?

No; for a 2-dimensional analogy, imagine a flat blackboard that goes on forever.

 

[B.C.]

Consider a two dimensional analogy. Take a sheet of paper, draw a triangle on it. Now roll it into a cylinder. The paper didn't have to stretch, so the sum of the angles of the triangle is still 180 degrees. The two dimensional beings living on the paper can't really visualize the operation, even though they may have taken topology at their two dimensional university. Now try to join the ends of the cylinder. We can't do that in our three dimensional space without the paper stretching and the angles no longer summing to 180 degrees.

But if we can go into a fourth spatial dimension, we can join the ends into a torus without stretching. Unlike a three dimensional donut, this torus is flat everywhere. And yet it is finite.

Now repeat the above, starting with a three dimensional solid cube...

 

[Chalnoth]

But if we can go into a fourth spatial dimension, we can join the ends into a torus without stretching. Unlike a three dimensional donut, this torus is flat everywhere. And yet it is finite.

A little bit of nitpicking here: actually, it doesn't matter how many dimensions you have. The important point is that torus itself has no net curvature, and so it is possible to think of an idealized torus without any. There's no reason why our universe cannot have this kind of shape.