Monday, July 16, 2007

75-year-old mystery of Möbius strip solved!

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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 (infinity) in three-dimensional form.

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.

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 The duo solved the problem using a set of unpublished 20-year-old equations.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.

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. 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. Link

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