Just to try to clarify the quotient construction, you start with a square tiling, then draw a diagonal across two squares, then an orthogonal diagonal across three squares. These are two of the edges of a rectangle. Complete the rectangle, and cut it out to get a rectangular strip, which you then glue to itself to make a torus. The vertices and edges are carried over from the tiling. 

 Another Construction is start with the rectangle with vertices (0,0) (0,2) (3,0) and (3,2). Cut it out and glue it together as  torus. Now put vertices on each point (x,y) where x is 0, 1, or 2, and  y is 0 or 2, OR where x is 1/2, 3/2, or 5,2, and y is 1/2, or 3/2. Now draw edges between vertices where the coordinates each differ by 1/2. The faces are the squares regions bounded by the edges. 173.227.48.5