Möbius Mysteries

Over a hundred years ago, a German mathematician, August Möbius, discovered that strange things happen if you twist a strip of paper before you tape its ends together to make a loop. Perhaps you have already discovered what happens when you cut down the middle of the length of the loop (see illustration). If you haven’t, guess what you think will happen, then make a Möbius strip and cut it lengthwise.

Now try twisting the strip twice (or three times!) before taping the loop closed and cutting. Then try cutting the loops from one third of the way from an edge, instead of halfway. What happens when you try cutting one fourth of the way from an edge?

If you draw a line along the middle of the loop instead of cutting it, you will find that you have come back to where you started only after your line goes on both the “inside” and the “outside” of the loop.