The Möbius strip is a great little impossibility, something that refuses to vanish no matter how long you stare at it. Perhaps you thought there couldn’t be an object with one surface and one edge? Guess again.
If you’re traveling on this surface, you will end up back where you started, but not before crossing your starting position while flipped over. Allow me to illustrate this with Wikipedia’s best gif:
Notice that the side the crab’s big claw is on flips. The crab is walking around underneath its original starting position, even though it’s still on a continuous surface.
Once you get over how profoundly wrong something like this feels, your next thought may be, “What can I do with this?” The answers are limited, and weird: conveyor belts that wear evenly on both sides, stage magic, and typewriter ribbons:
My favorite, however, is comic strips. XKCD did a brief one (you can read it here), but, for a true exploration of the form, watch “Wind and Mr. Ug,” below.