At Last, Mathematically Modeling a Möbius Strip's Shape

July 30, 2007

Discovered about 150 years ago, the Möbius strip has long resisted definition: When one end of a rectangular strip of paper is twisted through 180 degrees then joined to the other, what characteristic three-dimensional shape does the resulting one-sided, one-edged strip adopt?

Now, Eugene L. Starostin and Gert van der Heijden, of University College London, have finally come up with the answer. The duo solved the problem by making use of unpublished 20-year-old mathematics.

"If you try to write out equations for the shape of the strip without these tools it's a formidable task," Starostin told Nature. By using the rediscovered equations, however, the two researchers showed that any strip's shape depends on the length and width of the rectangle from which it is made.

A paper describing the work, "The Shape of Möbius Strip," appears in Nature Materials.

"This is the first application of this mathematical theory," Starostin said. Other scientists, it turns out, might find that the mathematical model has applications in their fields.

"The equations apply to any rectangular strip that twists and bends," mathematician John H. Maddocks, of the Swiss Federal Institute of Technology in Lausanne, told Nature. For instance, he said, the mathematics might be useful for carbon nanotubes, which are made from sheets of carbon. Or the same approach could be applied to understanding the shapes of molecules or explaining how telephone cords coil.

For his part, Starostin has been inspired to set his sights beyond the Möbius strip. "The same theory can be used to describe non-rectangular shapes," he said, such as trying to model the shape of lettuce leaves or chemical films. "We also hope this will help us understand crumpling," he added.

The Möbius strip has a long history of serving as inspiration for artists, architects, poets, engineers — and even roller-coaster designers. Conveyor belts, as it happens, can be manufactured as Möbius strips, because if the entire area of a belt receives the same amount of wear, it lasts longer. It's also a way of doubling the playing time of recording tape.

Source: Nature, July 15, 2007; Nature Materials, July 15, 2007