# Understanding the universe as a 3-dimensional surface

When thinking about the shape of the universe in my story I have been looking into it being a 3-manifold. As I understand it, space would be a 3-dimensional surface and for a small scale observer space would seem flat but if somehow it could be viewed from outside its shape would be curved.

Something I am not able to visualise or probably that I don't understand is if there is a thickness which would be a much shorter length than moving in other directions? So if you moved across the circumference of the surface/shape you would either return to where you started or move in a seemingly endless path looping around the shape but if you moved at 90 degrees to the circumference, would it be a much shorter distance to travel and if so, what would happen there? Would space curve back on itself?

Two of the easiest topologies to imagine this surface on is a hollow sphere and a torus, in this image the surface of the torus would be 3-dimensional space time and the grey cross section area is the depth or thickness of space which I am wondering if it would be a much shorter distance to reach than taking other paths and what would happen to space when you reach there?

Edits have been made to the question to correct parts which were mentioned in the comments.

• "So if you moved across the surface/shape you would either return to where you started or move in a seemingly endless path looping around the shape . . ." I don't follow this 100%. Maybe put a picture. Commented Mar 31, 2022 at 12:59
• "would space curve so that there is no "outside" of the space that is visible or accessible." I don't follow this either. Commented Mar 31, 2022 at 13:00
• A surface by definition has no depth. You may be confusing it with a sheet of paper, but a sheet of paper is actually a three-dimensional object, it just has a much smaller thickness than its width and length. (And, while you can view spacetime as a 4-dimensional manifold embedded in some higher dimensionality manifold, such a view is neither required nor useful in any way. For all the practical purposes of physics, the 4-dimensional spacetime is not embedded in 5-, 6- or higher-dimension space, and such an embedding won't simplify physics at all; in fact, it would complicate it.) Commented Mar 31, 2022 at 13:03
• A simple example of how "surface depth" might have no meaning is the manifold of heights and weights of people. Every point on the manifold is a pair of numbers (height, weight). All the possibilities knit together to form a 2d surface. But there is no perpendicular direction to this manifold. Commented Mar 31, 2022 at 13:06
• @AlexP I not fully following sorry. I have heard people saying the surface is embedded, which I didn't understand but in my visualisation I don't see or need other dimensions the curved space has that shape but nothing outside matters or exists.
– user94655
Commented Mar 31, 2022 at 13:12

If space really is a 3-dimensional manifold (which is what most theories of physics assume), then its "thickness," in the sense that you're talking about, is exactly 0. Even if we assume that there's some direction which is perpendicular to all three of our ordinary everyday dimensions, it's impossible to move in that direction, because the amount of distance available in that direction is 0.

According to Wikipedia, string theory may predict that there are more than 3 spatial dimensions, and Wikipedia mentions two possibilities for how this might work. One of the possibilities is called "compactification," and my understanding of this is that space does have a thickness, but the thickness is extremely tiny (maybe a few picometers, I dunno). These dimensions would curve back on themselves, so if you were to move in the direction of the thickness of space, you'd quickly end up back where you started.

The other possibility that Wikipedia mentions is "that the observable universe is a four-dimensional subspace of a higher dimensional space." In other words, the observable universe is a thin slice of something bigger, and I don't know if the thickness of that slice is 0, or something greater than 0.

• zero or close to, thickness sounds crazy and confusing lol, You mentioning string theory made me think of Brane theory, that also says it is very thin although there are thick branes but so far I haven't found out how thick they are. Having zero or close to, thickness makes me think of the Holographic principle or how we view the CMB.
– user94655
Commented Mar 31, 2022 at 16:37
• Well, having zero thickness is the simplest possible situation, because it means that the universe really is 3-dimensional, just like it seems to be. Having non-zero thickness would mean that the universe has at least one hidden dimension that hasn't been discovered yet, and I would call that pretty "crazy and confusing." Commented Mar 31, 2022 at 20:46
• I still don't really understand zero thickness being 3-dimensional, for holographic principle all information encoded onto a 2d surface kind of makes sense but something being 2d when viewed from outside but is actually 3d and infinite inside is hard to get my head around.
– user94655
Commented Mar 31, 2022 at 21:17
• I don't think anyone fully understands the holographic principle, not even theoretical physicists! So if I were you, I wouldn't worry about the holographic principle at all. As for why zero thickness would correspond to being 3-dimensional... well, let me try to explain it with an analogy. Is a sheet of paper (a real sheet of paper) 2-dimensional or 3-dimensional? (cont'd) Commented Mar 31, 2022 at 22:11
• At first glance, paper kind of seems like it's 2-dimensional. However, paper actually has a nonzero thickness, which means that it's really 3-dimensional after all. But if paper really did have exactly zero thickness, that would make it truly 2-dimensional. The thickness of paper provides an extra dimension, but if it didn't have that thickness, then it'd just be 2-dimensional. We can apply the same reasoning to space, but we have to add 1 to each number. (cont'd) Commented Mar 31, 2022 at 22:13

Folding Space

In the Sci-Fi Business, What you are talking about is called folding space. This is a type of mumbo-jumbo where we explain the existence of wormholes or faster than light travel by pretending the universe (three or four dimensional) exists in some larger space, the same way a sheet of paper exists in three-dimensional space.

Even though there is a speed limit on the paper -- hence a minimum time to travel from one corner to the other -- you can zip from one corner to the other if you have a magic sci-fi engine that bends the sheet to put the corners on top of each other. To creatures living on the sheet with no conception of the perpendicular direction, this would look like some sort of wormhole.

Of course this is all scifi mumbo-jumbo. In General Relativity the universe and its matter is described as a four dimensional manifold. There is a whole field of maths called differential geometry, which centres around how to refer to such objects using properties inherent in the manifold itself, rather than them existing in some larger space. Under this mathematical framework it doesn't even make sense to move in the perpendicular direction.

You don't even need an ambient dimension to describe wormholes. A universe with a wormhols is just a different type of 4d shape to a universe without a manifold. The difference is there is a tunnel in one and not the other. We don't need to suppose a higher ambient dimension to describe the tunnel.

• I am not talking about wormholes or manipulating a manifold, the topology of the universe in these theories is a manifold, like a sphere, torus or many other types with the surface being space.
– user94655
Commented Mar 31, 2022 at 13:00
• You are talking about a "3 dimensional curved surface". The mathematical version of that is just a 3-dimensional manifold. Commented Mar 31, 2022 at 13:03
• Going off yours and AlexP's comments above I still don't understand how to visualise that without comparing it to a hollow sphere or other shape?
– user94655
Commented Mar 31, 2022 at 13:09
• @JarredJones There is nothing wrong with imagining it as a hollow shape. For example a 2-d sphere with matter moving around it, and the matter causes bulges in the surface. You just have to keep in mind the space inside or outside the sphere has no physical meaning. Commented Mar 31, 2022 at 13:13
• But then comparing it to a 2d sphere do we not have that "thickness" issue of it being a shorter distance than across the sphere, or have I missed something still?
– user94655
Commented Mar 31, 2022 at 13:18