Life on a 4-Dimensional Doughnut

Sometimes a simple idea or assumption can lead to an interesting chain of thoughts. I've just watched an educational prime-time program about maths and they ended the episode with one of these ideas, deducted from the maths they were talking about earlier. The program was about dimensions, in particular the 4th dimension.
Imagine you are playing a computer game on your screen. An interesting feature of these games often is that you can walk off one side of the screen and you appear again on the opposite one, e.g. leaving the screen on the left and appearing again on the right. We normally don't think about this very much and just continue with the game. But what does that mean in terms of surface structure if this virtual, 2-dimensional world? How does the world have to be shaped to allow for that to happen?
An intuitive answer of many people would be that it has to have the shape of a sphere. Let's imagine our small 2D walker would stand on a sphere. He could go in any direction and, by walking straight, would reach the point he started from again from exactly the other side. So far so good. But now, think about distances travelled by our small guy. On a sphere, no matter which direction he chooses, the distance to the point of origin is always the same. How about the distances he has to travel in our 2-dimensional world? Even if we imagine the screen to be square, the distance is different for different starting angles (imagine going through one of the corners to appear on the opposite one). So it can't be a sphere!
Let's shape the 2-dimensional world in order to have the desired features. Think of the screen as a piece of paper of the same size. To be able to get to the bottom end by going "off paper" on the top end, we have to bend the paper to form a cylinder. By connecting top and bottom side we enabled our little walker to go back to his point of origin when walking north or south. Now we have to connect left and right by forming a doughnut shape (or more mathematical: a torus), voila! A never ending world for our little wanderer.
Now what have we done? From the viewpoint of our (still) 2-dimensional wanderer, this is a normal 2-dimensional landscape. He just knows that if he walks straight on in his perfectly flat world, he comes back to where he started. Remember: He never experiences up or down (therefore he never thinks of it as a flat world! What else could there be?).
What we (as 3D people) have done is to shape the 2-dimensional world in the 3rd dimension to a 3-dimensional object to do, what the small guy can only think of as magic.

Now let him walk his new world for a while and think about what the 4th dimension could be. To answer this question we can look at the dimensions we already know. To get from dimension 0 (a dot) to the 1st dimension, we can imagine two dots and connect them. Now I can get from one dot to the other by walking along the 1st dimension, passing many other dots on the way. We can do the same to get from the 1st to the 2nd dimension: Every 1-dimensional object is a line. If we draw two lines (imagine parallel ones for simplicity), we can connect them by drawing other lines in between till we filled the space. All these lines are in a 2-dimensional landscape of 1-dimensional lines. The step to the 3rd dimension is equally easy by connecting 2 planes by other planes. If you need a visual aid, you can watch this movie and stop when your head starts hurting (usually around dimension 6 ;-) ).

Suddenly the step from the 3rd to the 4th dimension is easy: We just take two 3-dimensional objects (e.g. yourself) and connect them by other 3D objects going slowly along a line of objects. Think about that for a while and you will see that this is what we normally refer to as time. Just imagine a red cube and a yellow cube and imagine a line of cubes in between slowly changing colour. We can see that all the time in movies! So a 3D-movie as a whole is just a 4-dimensional object, a dot in the 4th dimension, connecting two sets of 3D objects (start and end set of the movie...and before anybody destroys this nice metaphor in a comment, no cuts are allowed!)

Let's go back to where we started with our walker: Now imagine yourself being the wanderer. You walk through a 3-dimensional space. What would it mean to "walk-off the screen"? The space in which we walk a straight line has to be bent in the 4th dimension. From what we did with the piece of paper, this sounds easy! Then we can walk straight and come back to the place we started from. Remember, you would still be walking in the 3rd dimension just like the 2D guy who never left the 2nd dimension on his journeys. (we neglect that it would take time for us to walk there, therefore also moving in the 4th, just imagine you can walk really fast!). What else does that mean?
Some people think that our universe has a toroidal shape (i.e a gigantic doughnut) in the 4th dimension (whatever that looks like). Since light travels in a straight line, if you look really far into the universe, you should be able to see yourself standing there, looking away! But wait, travel needs time (what we've neglected before)! So e.g. the picture of Proxima Centauri, the closest star to our solar system, is what it looked like 4.3 years ago. What does that mean in a toroidal universe? One of the stars we can see is our own sun from millions of years ago? What happens if you travel there? If we could travel faster than light, could be arrive on our own planet in the past? Does "beaming" somebody from point A to point B just mean to bend our dimension so that these to points touch through the 4th and push the guy a bit?

So many questions, a long post, and just because I watched a program on TV. And so many more questions to think about.....