Riemannian geometry is the most influential non-Euclidean geometry in physics through Ein-stein's General Relativity. But the Riemannian space is not the most general non-Euclidean geometry, since it does not include torsion. Torsion is the result of an asymmetry of connection coefficients with respect to the swapping of indices. Different attempts have been done to take into account the torsion in gravitation. The very first attempt was due to Cartan just few years after the elaboration of General Relativity. In this attempt known as Einstein-Cartan theory, space includes curvature and torsion at microscopic level where, the physical origin of curvature is the mass while the physical origin of torsion is the spin of particles. But in lack of exprimental evidences, several other attempts have been done in this sence. In one of the lastest attempts, gravitational waves are considered to be a possible cause of torsion. In this article after defining the torsion in a non-Riemannian space, we examine geometrical effects of torsion via breaking of parallelograms and the observable effect of this phenomenon in physics through the special case of a typical binary system of stars as source of gravitational waves. AMS subject Classification 2010: 53B50 53Z05