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
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,
of straight across.