Random demos

solution on a torus

the torus is represented by the rectangle, with its top and bottom edges identified, and its left and right vertical edges identified

(if it helps, picture first pushing top and bottom away from you while rolling into a horizontal tube, with those edges meeting and sealing at the back. then make the tube long and flexible, and curve with the midsection coking towards you and the circular ends of the tube pushed back from you on a curve, forming the doughnut shape when the circular openings meet at the back and seal)