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I am trying to wrap my head around what a flat universe means; Trying to imagine why angles of triangles in spherical volumes are different.

Because my brain can't do that well and I became a bit annoyed by it, I started to think about a hypothetical circular one dimensional universe (a circle) - placing myself on the circle.

Without access to a "detached" view from the second dimension, just being able to look back and forward, I don't see any possible way to determine whether my universe is circular or not - apart of seeing my own back - if light already traveled around the whole circle.

So, thinking about two dimensions (a sphere, I guess), I still can't imagine to see any kind of circle, curvature or special angle, because I imagine the curves would lift into a "detached" third dimension which is not there from my point of view?

I don't understand how one would see curvature without access to a detached view of the higher dimension, could someone relieve me from this headache? - Does a flat universe actually mean that it is non-circular?

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The simplest "flat" universe is infinitely large, and hence "non-circular". But there do in fact exist flat universes (I mean, hypothetically) that are flat but finite. More on this below.

Flat universes

A flat universe is a universe where your good old high-school math works as expected: the angles of a triangle sum up to $\pi$ radians (180 degrees), the circumference of a circle is $\pi$ times its diameter, parallel lines never meet, etc. In a 2D analogy, this corresponds to the geometry of a tabletop; hence the name "flat".

Closed and open universes

But "flat" isn't the only possible global geometry — or topology — of space. You may, in principle, imagine any crazy geometry you like. However, we like to believe that the Universe is homogeneous and isotropic, i.e. that it looks the same everywhere and in all directions. Indeed, it seems to be on sufficiently large scales. If this is true, then there are only two other possibilities, namely "closed" (or "spherical") and "open" (or "hyperbolic").

Non-flat geometries are difficult to visualize in 3D, but the 2D analogy of a closed universe is that of a sphere, like the surface of a ball. On the 2D spherical surface, the angles of a triangle sum up to more than 180º, the circumference of a circle is less than $\pi$ times its diameter, and lines that start out parallel eventually meet. Moreover, if you travel far enough, you will get back to where you started.

The same is true in 3D spherical space. You may even, in principle, see your own back, although that would require either 1) that you start looking exactly at the Big Bang, in which case the light from your back will reach exactly when the universe collapses in a Big Crunch, or 2) that you have a somewhat fine-tuned amount of dark energy to slow down the collapse, without making it expand too fast so the photons never reach you.

Flat, but finite universe

As commented by zibadawa timmy, a flat universe doesn't necessarily imply that it be infinite. This is only true for the simplest case, but in fact geometry allows for 10 different flat but finite global topologies. In such a universe, traveling far enough you'd get back to where you started. The simplest of these is the 3-torus which can be thought of as a box with periodic boundaries; if you exit one face, you enter through the opposite face.

We don't know if our Universe is simple and infinitely large, or if it is less trivial and finite. One way to test this is to look for multiple images of the same objects in the sky. Most attempts, I think, have looked for repeating patterns in the cosmic microwave background (see e.g. Aurich et al. 2021), but to my knowledge none of these have returned any convincing results.

Measuring the curvature in theory

Visualizing the 2D sphere is easy when it's embedded in a 3D space. But to describe how large its curvature is, you actually don't need the embedding 3D. You can just do a geometric measurement and compare your result to what you'd expect in flat space.

The quantity usually used to describe the magnitude of the curvature is $K$, which for a 2D sphere equals $1/R^2$ where $R$ is the radius of the sphere. If you draw a triangle of area $A$, and measure its three angles $\theta_{1,2,3}$, then the curvature is given by $$ K = \frac{\sum_i \! \theta_i - \pi}{A}. $$ This formula is valid for spaces of any numbers of dimensions (higher than 1D).

For instance, consider a triangle going from Ecuador, to Gabon, to the North Pole, and back to Ecuador. In this case, the sum of the angles is 270º, or $3\pi/2$, and the area of the triangle is $1/8$ of Earth's surface area, i.e. $4\pi R_\mathrm{Earth}^2/8$. Hence, the curvature is $$ K = \frac{3\pi/2 - \pi}{4\pi R_\mathrm{Earth}^2 / 8} = \frac{1}{R_\mathrm{Earth}^2}. $$

There are several other mathematical ways to measure the curvature. For instance, you can parallel transport a vector along two difference routes; only in a flat space will they end up pointing in the same direction. In the example above, transporting a vector pointing from Ecuador to Gabon along the shortest path (red path in the figure below), and transporting it via the North Pole (green path), gives a difference of 90º.

Earth

So, to see the curvature of a 2D space in a higher-dimensional flat space requires that this space be 3D. But as described above, seeing the curvature is not required in order to measure it.

Furthermore, to see the curvature of a 3D space in a higher-dimensional flat space, actually 4D wouldn't be enough — you would need 6 dimensions. And to see a 4D space (e.g. spacetime), you'd need 10 dimensions. But just as in the 2D case, there's no need for it, and to my knowledge (but please correct me if I'm wrong) no-one claims the existence of such higher dimensions.

Measuring the curvature in practice

In practice, this it not how we measure the curvature of our Universe. There are basically two way to do this:

One is to measure the angle of something you know the size of, and compare to the angle you'd have in a flat universe. For instance, theory tells us how large the patterns in the CMB should be, and measuring the actual angles then puts constraints on the topology of the Universe

The other is to measure the energy density of the Universe, because the Universe should be open, flat, or closed if the total density is smaller than, equal to, or greater than a certain critical density. This can be done in several different ways, e.g. by observing supernova brightnesses, the CMB, the large-scale structure of the Universe, gravitationally lensed galaxy clusters, etc.

Luckily, the various more or less independent methods return roughly the same answer. And this answer seems to imply that our Universe is very close to flat; within the margins of the errors, it is consistent with being exactly flat (Planck Collaboration et al. 2018). Hence, for most practical purposes, we may regard the Universe as flat. There was, however, recently a paper by di Valentino et al. (2019) claiming evidence for a positively curved (i.e. closed) Universe. But they considered only one out of Planck's many data sets, so it seems to me a bit like cherry-picking.

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  • $\begingroup$ Thank you so much for this great answer, there is a lot of interesting information in it! Because of your answer I realized that I was imaging more of a grid/cube (which in my silly mind was distorted to look like a sphere in 2D) than an actual sphere, where I believe the interior angles of triangles still add up to 180 degrees. I'd love to read details on the measuring in practice if you have some material about it! $\endgroup$
    – Julius
    Jan 8, 2020 at 8:55
  • $\begingroup$ @Julius See my update in the last section :) $\endgroup$
    – pela
    Jan 8, 2020 at 13:24
  • $\begingroup$ Is a flat torus not flat in the cosmological sense? $\endgroup$ Jan 12, 2020 at 7:34
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    $\begingroup$ @zibadawatimmy Better late than never! Thanks for the comment, you're right, and I edited my answer to describe this. $\endgroup$
    – pela
    May 3 at 8:39

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