Life on a torus

On a wrap-around screen of width $W$ and height $H$, a point has coordinates $(x, y)$ with $0 \le x < W$ and $0 \le y < H$, and moving past an edge means counting on modulo $W$ or $H$ — clock arithmetic in two directions at once.

Gluing the left and right edges of a rectangle gives a cylinder. Gluing the two ends of the cylinder gives a torus: the surface of a doughnut. So the video-game world is a torus. A walker on it can go "around the hole" in two different ways — across the screen or up the screen — and neither loop can be shrunk to a point. Counting how many times a loop goes around each way is the start of the fundamental group of the torus, $\mathbb Z \times \mathbb Z$.

Now change the rule: walking off the top brings you back at the bottom mirrored, left and right swapped. The walker comes home as their own mirror image — their left hand has become a right hand. That surface is the Klein bottle, and a surface where this can happen is called non-orientable. The Klein bottle cannot be built in ordinary three-dimensional space without passing through itself, but the rule for its edges describes it perfectly well.

Mathematicians build many surfaces this way, by gluing the edges of a polygon, and the classification of surfaces says that every closed surface arises like this. Whether our own universe might be finite but wrap around in some such way is a question cosmologists have tested against observations; so far no evidence for it has been found.