‘Endless ribbon’ mystery solved

24 July 2007

Mobius strip

Dr Eugene Starostin and Dr Gert van der Heijden (both from UCL Civil & Environmental Engineering) last week published the solution to a 75-year-old mystery.

The two academics have discovered how to predict the shape of a Möbius strip, the ‘endless ribbon’ which is obtained by taking a rectangular strip of paper, twisting one end through 180 degrees, and then joining the ends.

The shape takes its name from August Möbius, the German mathematician who presented his discovery of a 3D-shape with only one ‘side’ to the Academy of Sciences in Paris in 1858. The shape was rediscovered by artists and famously depicted by Escher.

The first papers that attempted to work out how to predict the 3D shape of an inextensible Möbius strip were published in 1930, but the problem has remained unresolved until now.

Dr Eugene Starostin

Dr Starostin and Dr van der Heijden realised that the shape can be described by a set of 20-year-old equations that have only been published online. Their letter to ‘Nature Materials’ demonstrates that these differential equations govern the shapes of elastic strips when they are at rest, and enable researchers to calculate their geometry.

Dr Gert van der Heijden

Möbius strips are not merely mathematical abstractions. Conveyor belts, recording tapes and rollercoasters are all manufactured in this shape, and chemists have now grown single crystals in the form of a Möbius strip.

The academics believe their methods can be used to model ‘crumpled’ shapes that are not based on rectangular strips, such as screwed-up paper, the drape of fabrics and leaves.

“This is the first non-trivial application of this mathematical theory,” said Dr Starostin. “It could prove to be useful to other research communities, such as mechanics and physics.”

To find out more, follow the links at the top of this article.

Images: Dr Eugene Starostin (left) and Dr Gert van der Heijden