Shaping Up a Möbius Strip

Mathematicians finally calculate the shape of Escher's muse


Long known as curious mathematical objects lacking a separate "inside" and "outside," Möbius strips have also captured the imagination of artists like M. C. Escher, whose painting Möbius Strip II shows ants in a never-ending crawl on this curious surface. Easily made by twisting a strip of paper and gluing the two ends together, it is an object that only has a single surface and a single edge; Escher's ants crawling on the strip traverse all of its surface area without ever crossing an edge.

Nearly 150 years after Möbius strips were discovered, scientists at University College London (UCL) report in Nature Materials that they can calculate the exact shape of this odd object if given its aspect ratio (the ratio of width to length) along with the elastic properties of the material from which it is made. Apart from their purely mathematical significance, Möbius strips are sometimes used in machinery to transmit power between two pulleys using drive belts in which "both sides" wear equally. Despite their long history, however, no one could predict a priori what one of these strips would look like if you were to make it from, say, a three-inch wide, 20-inch long sheet of transparent plastic. The UCL scientists not only solved that mystery, but they also figured out the maximum width of such a strip given the length, putting to rest a question first asked more than 80 years ago.

Their result is a set of differential equations that can be solved given the elastic properties of the material and the aspect ratio of the sheet. Using the very general principle of minimal energy (which explains, for example, that bending a steel rod is hard work because the bent rod has higher elastic energy than the straight one), scientists can solve those equations to predict the shape of a Möbius strip when it is at rest. In addition to the mathematical satisfaction of figuring out a long-standing problem, the study has also paved the way for scientists to analyze the structural properties of macromolecules and crystals grown in the form of Möbius strips, a process developed in 2002.

DIY Möbius strips

Making a Möbius strip

Take a letter-size sheet of paper, and cut a strip an inch wide from the long side. Draw a line along the center of the strip. Now give it half-a-twist, and glue the ends together. The diagonally opposite corners of the original strip should come together, while the two ends of the center line should meet.

How is this one-sided?

Try taking a felt pen or crayon and start coloring one side of the strip, and keep coloring without crossing any edges. You'll soon find that you've colored both sides of the original paper! Or, start coloring one edge of the strip and keep going; you'll end up coloring both edges without cutting any corners.

Now, using a scissors, cut along the center line. Did you expect what you get?

Examples of Möbius strips

Besides M.C. Escher's paintings Möbius strip I and Möbius strip II, the universal recycle symbol is a Möbius strip. The shape is also used in decorative jewelry and architectures.

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