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