Practical joke for a topologist

Here is a practical joke that you may want to play upon a fellow topologist.

1. When he is not looking, get hold of his jacket, and cut out from it a round patch, a couple of inches across:

   

2. Take a strip of cloth whose width R is equal to the radius of the removed patch, and whose length L is one half of the perimeter of that patch, that is, πR. Bring the two ends of the strip together, with a half-twist turn, and sew them together so as to make a Möbius strip:

   

3. Place the Möbius strip over the hole and sew the two edges together:

   

Then replace the jacket and wait for your friend to put it on. Enjoy his confusion as he suddenly and inexplicably finds himself unable to tell whether the jacket is inside-out or not.

Note that there is no need to use the same type or color of cloth for the Möbius strip, since topologists are notoriously unable to perceive those details. Indeed, the strip could have any size or shape; the dimensions given above are merely to make it easier to sew onto the hole.