Gives the quadrangulation of a twocomplex along maps from edgepaths from the twocomplex to paths in the quadragulation. The forWard EdgePath method turns every edge in the original twoComplex into an EdgePath of length two in the corresponding face in the quadrangulation. The backWard EdgePath method only works if the EdgePath in the quadrangulation starts and ends at pre existing vertices, or is a loop on a barycenter of length more than one. In the first case it returns a fixed end point homotopic path in the original TwoComplex. In the second case if returns a freely homotopic path in the original TwoComplex. In the first case the backWard EdgePath map considers edges in the quadragulation in Pairs. Because the pre existing vertices and the barycenters form a bipartite set in the quadrangulation, a consequtive pair of edges in the quadragulation (say u and v), starting from a (which is a pre existing vertex), passing through b (which is a barycenter) and ending at c (which is a pre existing vertex) is homotopic to a sequence of edges in the face corresponding to b. So we can take such a path. In the second case because the pre existing vertices and the barycenters form a bipartite set in the quadrangulation, shifting the basepoint makes the loop start and end at a pre existing vertex.