A good example of a topological structure is called the Möbius strip (also called the twisted cylinder). It was invented by German mathematician and astronomer August Ferdinand Möbius (1790–1868) in September 1858, although he did not publish his findings until 1865. The strip was also developed independently by German mathematician Johann Benedict Listing (1808-1882), who discovered it in July 1858. Although Listing published his findings in 1861, Möbius’s name is used for the strip.

This one-sided surface is easy to visualize: Take a long single strip of paper, give one of the two ends a twist (that gives the strip a half twist in its length, or a 180 degrees twist), then attach the two ends. Amazingly, you can verify that it has only one side—it is not possible to paint it with two colors. Even if you cut the strip in half (down the middle), you will still have a one sided-surface. In fact, ants would be able to walk on the Möbius strip’s single surface indefinitely, since there is no true edge in the direction of their movement.