If I were to stand on a flat torus, would I be able to see myself from behind?

Question

The title pretty much says it all : from any point on a flat torus, there is at least one geodesic from that point that goes back to it. And, by definition of a flat torus, that geodesic is a line. So, if I look toward the direction pointed by that geodesic, do I see myself from behind ?

If the anwser is no, why ? I have very little knowledge about how a flat torus can not be embedded in $\mathbb{R}^3$ with full ($C^2$) regularity, so I suppose that we need to be at least in $\mathbb{R}^4$ for the question to make sense, but then maybe it is hard/not relevant to talk about light trajectory (obviously I am not considering any relativistic effect whatsoever).

Answer 1

Yes: if you're "in" the torus and light rays travel along its geodesics, then we can think of the torus as a flat square with its top & bottom and left & right edges identified. In terms of what you can see, it's equivalent to being in an infinite plane with a copy of yourself at each integer point; you see many copies of yourself from behind.

Edit: I think the answer is no to the question you mean to ask. Note that (per Wikipedia) the flat torus (e.g. as realized by the standard embedding in $\Bbb R^4$) is flat in the sense that the surface of a cylinder is flat, i.e. it has zero Gaussian curvature. Of course you don't see yourself from behind when standing on a cylinder in $\Bbb R^3$, and more generally, if you're in a Euclidean space where light travels in straight lines, then it doesn't matter what you're standing on, you won't see yourself from behind.

Answer 2

You don't need $\mathbb R^4$ at all to address your questions. In fact the Euclidean plane itself will suffice to depict what you will see. Here's a picture showing a head looking up outward from the torus:

The point is that the flat torus may be abstractly described by starting with the unit cell, and then identifying each point on the left edge with the corresponding point on the right edge (and similarly for the top/bottom edges). So light beams heading horizontally outward at the left side of the unit cell will reappear heading horizontally inward at the right side.

Now imagine that head seen in profile, looking to the right. That head will clearly see the back of itself.