In the above figure, I have drawn a closed straight path which starts from A on the bottom right edge and then hits the middle of the front face at B. It continues from the middle of the back face (since the front face is glued to the back face) and finishes at the middle of the top left edge at a point which is glued to A (via the top and bottom face gluings as well as the left and right side face gluings).
There are only 10 Euclidean possibilities for the shape of a closed Euclidean universe -- namely, the 3-torus and nine simple variations on it, such as gluing together opposite faces with a quarter-turn or with a reflection, instead of straight across.
The same type of reasoning is used to try and create a map of the structure of the internet: