In four dimensions, a flat torus is an object that has zero curvature but still has closed geodesic curves. What this means is that if you try to measure geometry locally, you will find that it is perfectly Euclidean. Nevertheless, if you travel on a straight line, you'll eventually end up where you started. What would happen if you carried out the twin paradox in a universe with such a geometry? In the standard twin paradox, one of the twins experiences acceleration effects, so his frame is not inertial. But in a flat toroidal universe, he would always be in an inertial frame, since going around in a closed curve doesn't require any "turning". So what would be the results? Would there still be time dilation?