If you lived in a 4-torus, what would the doughnut hole look like from the inside?

Question

I'm not just curious; it refers to general relativity.

Specifically, would the hole in the torus' center look to us like a sphere, one you cannot enter because you always slip across the side and go around it instead of through?

Answer 1

A person living in a torus can't see the hole. Internally, a two-dimensional person living in a 2-torus can see that the geometry is flat, and that if they travel far enough in one direction they come back to a place near their starting point. But the “hole” is invisible, because that's entirely a property of the way the torus is immersed in 3-space.

Video games often have a toroidal geometry, in which that game space is a rectangle where going off one edge returns one to the opposite edge:

so one might ask whether the little spaceship in the game can see the “hole” in the donut. And the answer is that while the surface of a donut can model such a space, you don't need the donut, all you need is the rectangle with the connected edges. In this model of the torus, there is literally no hole anywhere to be found.

For a 4-person in a 4-torus, the answer is the same. The torus itself is flat. The “hole”, if there is one, is a property of the way the torus resides in the enclosing space, if it does reside in an enclosing space. But whether it does or not, those details are completely invisible to a person in the torus.

Answer 2

There is not necessarily any "middle" of a four-torus, nor is there necessarily any "hole" in any place you might call the "middle".

When we try to visualize space in cases where the space is not Euclidean, there's a strong inclination to try to do this by embedding the space in a Euclidean space of higher dimensions. But this is often a distortion of the actual structure of the space.

The old Asteroids game from the early days of arcade video games is a good example.

The asteroids in this game traveled at constant velocity through a two-dimensional toroidal space. That is, when an asteroid went off the bottom of the screen it reappeared at the top, and when it went off the right edge it reappeared on the left edge. In each case it reappeared with the exact same speed and direction.

We call the space in which the Asteroids game lives "toroidal" because one way to visualize its topology is to imagine that we put a photograph of the game screen on a rubber sheet and glue the left edge to the right edge to form a cylinder. That lets objects cross the left or right edge of the screen and come back in on the opposite edge. Then we can bend this cylinder into the shape of a torus to connect the top edge of the screen to the bottom edge.

But now a velocity that was constant in the game is no longer constant in the 3-space in which the torus is embedded, because we had to stretch or compress the rubber sheet in order to bend it into the shape of a torus.

Even worse, we could just as well have glued the top edge to the bottom edge to make a cylinder that way and then glued the left to the right. The difference between these embeddings is that in one case, you can encircle the "hole" of the "donut" by traveling up the middle of the screen from bottom to top, but traveling from left to right makes a circle that doesn't go around the "hole"; whereas in the other case you encircle the "hole" of the "donut" by traveling across the screen from left to right, but going bottom to top makes a circle that doesn't go around the "hole".

That is one reason why one might not want to embed a space such as a 4-torus in another space at all. The embedding is likely to poorly model properties of the embedded space and is likely to create "properties" that the space doesn't really have (such as being able to "encircle" a "hole" in one direction but not in another direction).

And why should there be an embedding, in particular why should there be an embedding in a Euclidean space? The whole point of general relativity is that space is not necessarily Euclidean. The point is not that there's some higher-dimensional space in which our space can be embedded.

If we do not insist on an embedding, the "hole" is not only impossible for the people living in the non-Euclidean space to detect; it truly does not even exist at all (in the sense that we commonly speak of the hole in a doughnut in common non-mathematical speech).

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There are some other ways to look at "holes".

There is an embedding of the Asteroids topology in four-dimensional Euclidean space called the Clifford torus. This embedding doesn't require the surface to be "stretched" the way the three-dimensional embedding does. It also doesn't have the same kind of "hole" as the three-dimensional embedding does. Sure, you can project the Clifford torus onto three-dimensional space to get something that looks like a doughnut. But you can also just as easily project the Clifford torus to make another doughnut that's turned inside out from the original one: points in the "hole" in the first doughnut are now part of the doughnut itself (the part you would eat if it were a real doughnut) and points that were part of the first doughnut are all outside the new doughnut.

So is there a hole in the Clifford Torus?

If you consider the kind of "hole" counted by Betti numbers, as outlined in another answer (and if you're reading this and haven't yet upvoted that answer, I suggest you do so), the Clifford Torus has two "one-dimensional" holes. By the same account, the surface of a doughnut in three-dimensional space should have two of these holes as well. But where are they, exactly?

"Topology 101: The Hole Truth" identifies the two holes on a two-torus by displaying two distinct loops you can make on the torus, labeled $a$ and $b$. There is a $b$-loop that encircles what a non-mathematician would identify as the "hole in the doughnut" with nothing else inside that circle. There's no such $a$-loop. But due to the symmetry of the Clifford torus, there's just as much "hole" inside an $a$-loop as inside a $b$-loop.

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So there are at least two reasons it is problematic to ask about "the hole in the torus' center". One reason is that this presumes an embedding of the four-torus in some higher (presumably Euclidean) space, which I think is a parochial point of view: space looks locally Euclidean to us, so in reality whatever the shape of space is, we suppose (without justification) that it must somehow live in a Euclidean space that we would be able to see if only we ourselves were high-enough-dimensional beings. The other reason is that even if the four-torus were embedded in a higher-dimensional space, it should have multiple "holes" of various dimensions, and the "center" of any of these holes might not be as easily identifiable as the center of the "hole" that a non-mathematician identifies in a doughnut.

Answer 3

Some of the other answers say that there is no way for an inhabitant of the $4$-torus to "detect" the hole. This is not quite correct.

If the inhabitant of the $4$-torus can put a marker at some point and then follow a geodesic, then the existence of a hole can be detected as follows.

The inhabitant puts a marker at some point and then follows a geodesic, at some point in the future the marker will again be close enough to be visible. This will prove that there is a hole and that it is not flat space that is inhabited.

By doing this in multiple linearly independent directions, one can probably detect that it is not a cylinder either. Update: As @MJD has pointed out in the comments below there are other topologies to consider if we do not assume that the space is flat.

https://en.wikipedia.org/wiki/Linear_flow_on_the_torus

Answer 4

One way to "visualize" the 4-torus is by $$T^4=S^1\times S^1\times S^1\times S^1$$ This is an ordered set of four angles.

We have the notion of Betti numbers and we often say that "$b_n$ counts the number of $n$-dimensional holes." For $T^4$, it turns out that Betti numbers are $b_0=1,\ b_1=4,\ b_2=6,\ b_3=4,\ b_4=1$. The binomial sequence gives the pattern! Contrast this with the Betti numbers for $T^2$ which are $b_0=1,\ b_1=2,\ b_2=1$.

From the representation above of $T^4$ as the product of four circles, you can clearly see the four one-dimensional holes. How do we see the six two-dimensional holes? Select two of the four circles: there are six ways to do this. Then look at the 2-torus $S^1\times S^1$ formed by that pair.

Answer 5

The OP specifically asks about a 4-torus, still many of the answers are given with reference to a 2-dimensional person living in a 2-torus. It is claimed that a 2D person can not see the curved structure we associate with a torus (a donut-like shape). It seems the problem about "seeing the hole" is that it belongs to a higher order dimension. The OP doesn't specify what order of dimensions the person looking at (or for) the hole, belongs to, so it's hard to decide. If it's a 5-or-more-dimensional person, then he or she should be able to see the hole. An example in 2D is given with video game worlds, that exhibit the properties of 2-toruses. Again, the game is only playable in 3 dimensions. People say a lot about n-dimensional people living in n-dimensional spaces, that raises a few questions in my opinion.

One question: Can a 2-torus exist at all, unless embedded in 3-dimensional space? Even if embedded in 3D space, could the curvature overall be so smooth that it could be perceived as flat like in the video game by a 2D person living inside? For instance, the outer side of the torus has a longer circumference than the inner, leading to a world where one route around the world would be quite much longer than another.

Another thought: People talk about these imaginary 2D people as "flatlanders", and indeed it is easy to imagine such people as living on a flat world, where all is flat. But thinking a bit closer, we realise that they can not live "on" anything at all. They must be suspended in 2D space along with all other objects. If there actually were natural laws, like gravity, they would only work on the 2D plane. One might imagine a large 2D universe, resembling ours, with natural laws and stars,planets and everything. Then, the planets, viewed from the "top", our 3D privilege, would be flat discs. A planet like earth would be glowing hot in the center, and with people walking along the edge. They can not pass each other, however, unless by climbing over one another. High walls would be impossible to walk around, and the same problem applies to trees and animals, which could not pass each other. Me may conclude that life as we know it would not exactly thrive on this planet, and that higher order species would be implausible. But then again, nature seems to be creative in adapting itself to all sorts of conditions, so how would we know? Plants in general dont have much need to move about, also birds and insects can pass each other in the air. Also fish and sea-living organisms, one may imagine large flat seas, full of flat water. All in all, I imagine the "flatlander" perspective would be different than what we may think in the first place.