Scientists Say: Möbius strip

This mathematical oddity has only one surface

A Möbius strip is a loop with a half-twist in it. This gives the shape the strange property of having only one surface.

May 22, 2023 at 6:30 am

Möbius strip (noun, “MOH-bee-us strip”)

A Möbius strip is a loop with a half-twist in it. You can quickly make one using a long, rectangular piece of paper and some tape. Just bring the two ends of the paper strip together — but before taping them to each other, flip one end of the strip upside-down.

This loop may be easy to make. But the twist gives the shape a strange property: a Möbius strip has only one surface. To see how this works, draw a line down the center of a paper Möbius strip. Without ever picking up your pencil, you can draw a line that runs along parts of the loop facing inward, as well as those facing outward.

Here’s how to make your own Möbius strip at home. See how drawing a line on one “side” of a Möbius strip covers both the “inside” and the “outside” of the loop. This is because one end of the strip is flipped over before the two ends are connected. As a result, the end of one side of the strip is the beginning of the other side — so that the two sides form a single, continuous surface.

This is different than if you had a loop of paper with no twist in it. In that case, you would have to draw one line along the outside of the loop, pick up your pencil, and then draw another line along the inside of the loop.

Another strange property of a Möbius strip? If you cut your strip in half along a line down the center, you would not end up with two smaller Möbius strips. You would instead create a larger loop.

Two German mathematicians discovered the Möbius strip independently in the 19th century. One was August Ferdinand Möbius. The other was Johann Benedict Listing. Their discovery was foundational to the field of topology. That branch of math deals with the properties of shapes and surfaces.

Möbius strips have wide-ranging uses. For instance, they can be used to make conveyer belts or other machinery. Belts made with normal loops tend to wear out on one side but not the other. But with a Möbius strip, both “sides” of the belt are really the same side. So, the belt gets even wear on all its parts. This makes the belt last longer.

Möbius strips and the math related to them are also useful for scientists. For example, understanding such complex shapes can help researchers probe complex structures such as chemical compounds.