I'm fascinated by the animation of the chequered rectangle turning into a cylinder then into a torus, but would love to see the cylinder given a 180 degree twist before joining up its ends to form a kind of Mobius torus. The chequerboard pattern would presumably be even more distorted, and the ends would line up pale to pale and dark to dark squares presumably.

Come to think of it, can a standard Mobius band wrap anything?

Chris G