A New Twist on the Möbius Strip

16 July 2007 (All day)

Starostin and Van der Heijden

Kinky strips.
Regions of high energy density (red) show up in the kinks of wide Möbius bands.

Cut a thin strip from a piece of paper, twist it, and connect the two loose ends. You'll end up with a Möbius strip, a graceful bracelet that oddly has only one side, as you can easily demonstrate by running your finger around it. Now try the same thing with the much wider strip of paper. Why is it harder to connect the ends? Mathematicians now have a precise answer.

Although the general shape of the Möbius strip has been well understood by mathematicians and artists like M.C. Escher alike, no one had solved the mathematical equations that dictate its shape and specify where along the surface it curves and how sharply. The bending and twisting of the paper creates stresses that increase the energy stored within the strip. The equations, attempted as early as 1930, describe how the strip will arrange itself to minimize that energy. But the mathematical machinery didn't exist to solve them.

New calculations have solved the equations to produce simulated Möbius strips. And it can explain why wide bands make bad Möbius strips. The energy of twisting is greater in a thicker strip, which kinks wherever the material can't support the stress. The calculations also predict that thin bands curve more smoothly, a result the researchers call intuitive.

To solve the equations, mathematicians Eugene Starostin and G. H. M. Van der Heijden of the University College London turned to a 1989 theory that can solve certain families of differential equations, so-called Euler-Lagrange equations, but had never been applied to the Möbius problem. To the surprise of the authors and the mathematics community, the theory exactly predicted the shapes of differently proportioned Möbius strips, up to the critical limit where the strip flattens into an equal-sided triangle.

"This computation will be a classic," says mathematician John Maddocks of the Swiss Federal Institute of Technology in Lausanne. "What kid who's interested in science hasn't made one of these?" Maddocks wonders why it took researchers so long to tackle the problem with the appropriate tool, but adds that 18 years between the invention of the applicable theory and the solution is a blink of an eye "in mathematical time."

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