Mobius Strip: ‘Endless Ribbon’ Mystery Solved
Dr Eugene Starostin and Dr Gert van der Heijden (both from UCL Civil & Environmental Engineering) recently published the solution to a 75-year-old mystery.
A Möbius strip made with a piece of paper and tape. (Credit: David Benbennick / Courtesy of Wikimedia Commons)
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 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.
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.”
Nat Mater. 2007 Jul 15; [Epub ahead of print]
The shape of a Möbius strip.
Starostin EL, van der Heijden GH.
The Möbius strip, obtained by taking a rectangular strip of plastic or paper, twisting one end through 180( composite function), and then joining the ends, is the canonical example of a one-sided surface. Finding its characteristic developable shape has been an open problem ever since its first formulation in refs 1,2. Here we use the invariant variational bicomplex formalism to derive the first equilibrium equations for a wide developable strip undergoing large deformations, thereby giving the first non-trivial demonstration of the potential of this approach. We then formulate the boundary-value problem for the Möbius strip and solve it numerically. Solutions for increasing width show the formation of creases bounding nearly flat triangular regions, a feature also familiar from fabric draping and paper crumpling. This could give new insight into energy localization phenomena in unstretchable sheets, which might help to predict points of onset of tearing. It could also aid our understanding of the relationship between geometry and physical properties of nano- and microscopic Möbius strip structures.
PMID: 17632519 [PubMed – as supplied by publisher]