Möbius strip unravelled

Mathematicians solve 75-year-old mystery of infinite loop's shape.

Louis Buckley

The Möbius strip has inspired artists — such as Max Escher — as well as mathematicians.

M.C. Escher
Eugene Starostin's desk is littered with rectangular pieces of paper. He picks one up, twists it, and joins the two ends with a pin. The resulting shape has a beautiful simplicity to it — the mathematical symbol for infinity () in three-dimensional form. "Look," he says, as he traces his finger along its side, "whatever path you take, you always end up where you started."

Discovered independently by two German mathematicians in 1858 — but named after just one of them — the Möbius strip has beguiled artists, illuminated science lessons and stubbornly resisted definition.

Until now, that is. Starostin and his colleague Gert van der Heijden, both of University College London, have solved a conundrum that has perplexed mathematicians for more than 75 years — how to predict what three-dimensional form a Möbius strip will take.

The strip is made from what mathematicians call a 'developable' surface, which means it can be flattened without deforming its shape — unlike, say, a sphere.

When a developable surface is formed into a Möbius strip, it tries to return to a state of minimum stored elastic energy, like an elastic band springing back after being stretched.

But no one has been able to model what this final form will be. "The first papers looking at this problem were published in 1930," says Starostin. "It seems such a simple question — children can make these things — but ask the experts how to model this shape and we've had nothing."

Lost equations

The duo solved the problem using a set of unpublished 20-year-old equations. "If you try to write out equations for the shape of the strip without these tools it's a formidable task," says Starostin. "I tried it and it didn't work."

With the equations, the two researchers showed that the strip's shape depends on the length and width of the rectangle it is made from.

Starostin wants to alert other scientists to the existence of these forgotten mathematical tools. "This is the first application of this mathematical theory. Other communities, such as experts in mechanics, don't know of its existence."

Scientists in many different fields might find the model useful. "The equations apply to any rectangular strip that twists and bends," says John Maddocks, mathematician at the Swiss Federal Institute of Technology in Lausanne. "They might be useful for carbon nanotubes, for example, which are made of sheets of carbon."

The same approach could also be applied to understanding the shapes of biological molecules, or to explain why a telephone handset cord coils both to the left and to the right, says Maddocks. The work is published in Nature Materials1.

Sculpture and conveyor belts

Art and mathematics discovered the Möbius strip independently of one another, and in the same way — by playing with pieces of paper2. Many years after August Möbius presented his discovery to the Academy of Sciences in Paris, the Swiss artist Max Bill thought he had invented a new shape upon creating his 1936 sculpture, Endless Ribbon, designed to look like "flames rising from a fire".

Since then the Möbius strip has inspired numerous artists, architects, poets and even roller-coaster designers. Conveyor belts are manufactured as Möbius strips, because the entire area of the belt receives the same amount of wear, so it lasts longer. The same goes for recording tapes, as it doubles the playing time.

Starostin, however, has set his sights beyond Möbius strips. "The same theory can be used to describe non-rectangular shapes — for example, in trying to model the shape of lettuce leaves and also on chemical films. We also hope this will help us understand crumpling," he adds.

"I want to show you something," says Starostin, leaning forward in his chair. "Look at this." He points to a holly leaf on top of his computer monitor. "One of my targets is to work out the shape of this. Just look how complicated it is!"

References
  1. Starostin, E. L. & van der Heijden, G. H. M., et al. Nature Materials doi:10.1038/nmat1929 (2007).
  2. Emmer, M. Leonardo 13, 108-111 (1980). 

http://www.nature.com/news/2007/070709/full/070709-16.html