[floatrock]

> They get smaller as they get closer to the edge, but there's an infinitude of them, so you have a universe that is infinite from "inside",

> Our universe is not anti-de Sitter- it appears to be flat and does not pack away nicely into a bounded area like the Escher universe

How do you tell from the "infinite inside" whether it packs away nicely into a bounded area? Or is figuring that out the trick?

[antidesitter]

> How do you tell from the "infinite inside" whether it packs away nicely into a bounded area?

Roughly speaking, you measure how fast the volume of a ball grows with its radius [1]. If it grows faster than you’d expect in Euclidean space, you know you’re in a hyperbolic space.

You can also draw a big triangle and see whether the interior angles add up to less than 180 degrees [2].

[1] https://en.wikipedia.org/wiki/Scalar_curvature#Direct_geomet...

[2] https://en.wikipedia.org/wiki/Hyperbolic_triangle#Properties

[roywiggins]

And we've done exactly that using our cosmological models of the Big Bang:

https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.htm...

One side of the triangle is the width of Cosmic Microwave Background variations (calculated from models of the early universe); the other two sides are known from how far away the CMB appears to be, which is known from independent measures of the expansion of the universe. Some trigonometry will tell you what the interior angle that should make with the Earth, which you'll see as the angular width of the variations in the sky. You can compare those two numbers (the one by trigonometry and the one observed), and determine whether they are different enough to exclude a flat universe.

In other words, if someone holds an 1 foot ruler at a distance, you can use trigonometry to work out how far away it is by using the apparent size. But if you know how far away it is, and the apparent size disagrees with the trigonometry, then the shape of the universe must not be flat. The longer the ruler and the further away it is, the greater the deviation will appear (if there is one). The CMB is very far away indeed so it makes for a good ruler.

[hammock]

> And we've done exactly that using our cosmological models of the Big Bang

What's the result- flat or not?

[roywiggins]

Flat, as far as WMAP can tell!

https://wmap.gsfc.nasa.gov/universe/uni_shape.html

Of course the curvature could maybe be arbitrarily small, but... it's definitely very close to flat.

[jazzyjackson]

Hold up, so the flat-earthers just need to think a little bigger?

(Legit tho I've been trying to understand the 'shape' of the universe for a while now -- in the sense of, when I look at stars and galaxies in the sky above me, what direction am I looking as it relates to where things are in the universe relative to each other? As a kid I took Bill Nyes word for it: everything is on the surface of a balloon, expanding from the center. But do I ever see the far side of the balloon? Or is everywhere I look somehow constrained to the surface in every direction away from me, such that the 'other side of the balloon' is infinitely far away... but if I could see far enough, my own spot on the surface of this sphere would be visible to me, minus a few billion years... I want to understand this but I am very confused! This looks like an informative website so I'll keep reading ... thank you)

[roywiggins]

The balloon-surface model works if the universe is actually positively curved. It bends around in on itself and comes back. So you would be able to see yourself (eventually) after light made its way around the surface and back to you.

As far as anyone can tell, though, the universe is flat. So photons traveling outwards from you will never come back around- they'll just keep going.

I'm partial to the raisin bread model of the universe:

https://www.loc.gov/rr/scitech/mysteries/universe.html

It's a tad misleading: the raisin bread is, probably, infinite in extent. As a practical matter, we can only see a finite portion of the raisin bread. As we gaze more deeply into the sky, we're looking further into the raisin bread. Because light only travels at a finite speed, the further something is away, the older the image we see of it is. At a certain point, all we can see is the goopy bread batter that the raisin bread used to be: that's (sort of, kinda) the Cosmic Microwave Background: it's the oldest light in the universe, and we can't see anything older than it.

So the universe looks like a bunch of galaxies, with us at the center, and a sphere of microwave radiation at the edge. But every single observer sees a similar sphere around their exact location, so it's a kind of illusion.

[danbruc]

[...] whether it packs away nicely into a bounded area [...]

That is nothing you should worry about, with enough deformation you can make a map of any shape you like for any space. You must not conclude from looking at a map of the Earth using for example a Mercator projection that there are somewhere four straight edges meeting at right angles and where you can just fall off Earth. You also must not naively make conclusions about the relative sizes of things as they get heavily distorted. The same holds for this map of hyperbolic space, it is really misleading - in the space there is no boundary, there is no special point in the middle, it does not pack nicely into a circle in any meaningful way, ... Those are all just artifacts of the way this map is drawn and it has nothing to do at all with the underlying space.