16 July 2007

Moebius strip riddle solved at last

Agençe France-Presse
Scientists have cracked a 75-year-old riddle involving the Möbius strip, a mathematical phenomenon that has also become an art icon.
Moebius strip riddle solved at last

Loopy logic: A Möbius strip made of a material that changes colour with bending pressure. Places where the strip is most bent have the highest energy density; conversely, places that are flat and unstressed by a fold have the least energy density. Credit: Nature Materials/Starostin & van der Heijden

PARIS: Scientists have cracked a 75-year-old riddle involving the Möbius strip, a mathematical phenomenon that has also become an art icon.

Popularised by the Dutch artist M.C. Escher, a Möbius (or Moebius) strip entails taking a strip of paper or some other flexible material. You take one end of the strip, twist it through 180 degrees, and then tape it to the other end.

This creates a loop that has an intriguing quality – dazzlingly exploited by Escher – in that it only has one side.

See Escher’s Möbius Strip II (1963) from the M.C. Escher Foundation web site.

Mathematical conundrum

Since 1930, the Möbius strip has been a classic poser for experts in mechanics. The teaser is to resolve the strip algebraically – to explain its unusual shape in the form of an equation.

Now, in a study published in the journal Nature Materials that lyrically praises the strip for its “mathematical beauty,” Gert van der Heijden and Eugene Starostin of University College London, in England, present the solution.

What determines the strip’s shape is its differing areas of “energy density,” say the experts in non-linear dynamics.

“Energy density” means the stored, elastic energy that is contained in the strip as a result of the folding. Places where the strip is most bent have the highest energy density; conversely, places that are flat and unstressed by a fold have the least energy density.

If the width of the strip increases in proportion to its length, the zones of energy density also shift, which in term alters the shape, according to their equations. A wider strip, for instance, leads to nearly flat, “triangular” regions in the strip, a phenomenon that also happens when paper is crumpled.

Not just esoteric

The research may seem esoteric, but van der Heijden and Starostin believe it also has practical applications.
It could help predict points of tearing in fabrics and also be useful for pharmaceutical engineers who model the structure of new drugs.

“One of the classic problems in mechanics is to find the shape assumed by a Möbius strip – the famous band that is closed with a half-twist and which has the intriguing topological property that it only has one side,” said mathematician John H. Maddocks in an accompanying commentary that also appeared in Nature Materials.

“This abstract mathematical question, dating back to at least 1930, is also of practical scientific interest as single crystals in the form of a Möbius band have now been grown,” said Maddocks, of the Swiss Federal Institute of Technology in Lausanne, who was not involved in the study.

The Möbius strip was named after a German mathematician, August Ferdinand Möbius, who discovered it in 1858. Another German, Johann Benedict Listing, separately discovered it in the same year.

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