# Is our universe flat?

I heard our universe is flat. Then one question is puzzling in my mind. If our universe is really flat, why we measure distance from any point across in a spherical way. In other words, why we say, the comoving distance (radius) is now about 46.6 billion light years. If this radius is only treated for observable universe, then how does science find that the universe is flat knowing nothing beyond the limit of the universe.

There are some questions in SO, however, they do not assimilate what I intended to know. In this question, for instance, OP asked about whats the shape of the universe. And also there are some terms related to black holes, 3D shape, ball etc. Apart from those complexity, my intention is simply to be clarified, why we call our universe flat? And by word flat, what I meant "having a level surface; without raised areas or indentations". Am i missing any alrernative meaning of flat here?

• Possible duplicate of What is the physical, geometric shape of the universe? – Sir Cumference Mar 19 '17 at 19:55
• I don't think the question is a dupe of that one, where the OP knows that "flat" is the 2D analogy word. This question requires a more basic explanation of the terminology, I think, and I actually can't seem to find one here that addresses exactly that. <blatant self-promotion>This answer, however, does to some extent.</blatant self-promotion> – pela Mar 19 '17 at 22:18

And also there are some terms related to black holes, 3D shape, ball etc. Apart from those complexity, my intention is simply to be clarified, why we call our universe flat? And by word flat, what I meant "having a level surface; without raised areas or indentations".

Imho in order to accept the answer, you have to realize the contradiction in the sentence above: You mean level surface, but we call the universe flat :) "Flat" in general relativity does not refer to "having a level surface; without raised areas or indentations". Please accept this first.

Now let's think of a ball (yes, a ball; it's not complexity). Would you call its surface flat or curved? I'd call it curved. Let's think of a pan. I'd call its surface flat.

Both the surface of a ball and a pan have two dimensions. The ball itself has three dimensions obviously, but its surface has two dimensions. I hope this is straightforward.

Now, everything described above about surfaces also applies to volumes. Volumes can be flat, and they can be curved. The issue is that humans cannot visualize things with three dimensions bending, so they cannot visualize a curved volume. But this is okay: We can still grasp the meaning of flat and curved, just by seeing it on surfaces, and understand that it can be extended to volumes.

However it's not just visualization and understanding: Flat and curved volumes (i.e. spaces) have very different physical and mathematical properties. A usual example is that in a curved space the sum of the angles in a triangle is not 180 degrees. There are also other properties and those enable astrophysicists to measure the actual curvature of the universe. I hope this was helpful.

Flat here means "has Euklidean geometry" (on large scales), as explained by Helen's answer. Observationally, the universe appears flat, i.e. any deviations are within the uncertainties. Of course, if the universe is infinite, no statements on the whole universe can ever be made, but only on our local observable patch of it.

The aspect of the universe described as "flat" is the spacetime curvature. In General Relativity gravity isn't treated as a force but as a curvature of the spacetime manifold. At large scales the universe is expanding: negative curvature like the top of a hill, where things roll downhill away from each other. But mass causes a positive curvature like a bowl, where things roll together, toward the bottom of the bowl. On average the two regimes of curvature appear to be balanced - i.e. the universe is, on average, flat.

• Your analogy of negative curvature isn't quite right. A hill and a bowl both have positive curvature, like a ball. In all three examples a triangle would have sum of angles >180º and parallel lines meet, which is characteristic for positive curvature. A better (though not perfect) example is that of a saddle, where a trianlge has <180º and parallel lines diverge. – pela Mar 19 '17 at 22:08