The equilibrium shape

of an elastic developable Möbius strip


Figure: Computed Möbius strips of aspect ratios 5π, 2π, π, 2π/3.

Colour codes the bending energy density (from violet for almost flat regions to red for highly bent).


The Möbius strip obtained by taking a rectangular strip of plastic or paper,

twisting one end through 180 degree, and then joining the ends,

is the canonical example of a one-sided surface.



As simple experimentation shows, a physical Möbius strip, when left to itself,

adopts a characteristic shape independent of the type of material (sufficiently stiff for gravity to be ignorable).

This shape is well described by a developable surface that minimises the deformation energy, which is entirely due to bending.



We assume that the material obeys Hooke's linear law for bending,

then the energy is proportional to the integral of the non-zero principal curvature squared over the surface of the strip,

which is taken to be an isometric embedding of a rectangle into 3D space.



The problem of finding the equilibrium shape of a narrow Möbius strip was first formulated in 1930 by M.Sadowsky who turned it into a 1D variational problem represented in a form that is invariant under Euclidean motions.

Later W.Wunderlich generalised this formulation to a strip of finite width,

 but the problem has remained open although geometrical constructions of developable Möbius strips have appeared.

We apply an invariant geometrical approach based on the variational bicomplex formalism to derive the first equilibrium equations for a finite-width developable strip thereby giving the first non-trivial demonstration of the potential of this approach. The boundary-value problem for the M
öbius strip offers a fitting example for application of these equations.



Numerical solutions for increasing width-to-length  ratio show the formation of creases bounding nearly flat triangular regions,

a feature also familiar from fabric draping and paper crumpling.

This suggests that our approach could give new insight into energy localisation phenomena in unstretchable elastic sheets,

which for instance could help to predict points of onset of tearing.


E. L. Starostin & G. H. M. van der Heijden.  The shape of a Möbius strip.

Nature Materials 6 563-567 (2007) Published online 15July 2007. DOI:10.1038/nmat1929




Figure: Computed Möbius strip of aspect ratio 5π.

Colour shows the bending energy density (from violet for almost flat regions to red for highly bent).



·         J.H. Maddocks. Mathematics: Around the Möbius band. Nature Materials 6 547-548 (2007) Published online 15July 2007. DOI: 10.1038/nmat1960

·         Möbius strip unravelled. Published online 15 July 2007. DOI: 10.1038/news070709-16

·         Research Highlights: Möbius in equilibrium. Nature Physics 3 513 (2007) DOI: 10.1038/nphys694

·         A New Twist on the Möbius Strip. ScienceNow Daily News 16 July 2007

·         A Twist on the Möbius Band. Science News, week of July 28, 2007; Vol. 172, No. 4

·         Maths of Möbius strip finally solved.  New Scientist, issue 2613, 21 July 2007, p. 6

·         Shaping Up a Möbius Strip. Scientific July 17, 2007

·         Rätsel des Möbiusbands gelöst. 19. Juli 2007

·         Moebius strip riddle solved at last. Cosmos 16 July 2007

·         Rätsel der Endlos-Rennstrecke gelüftet. Sueddeutsche Zeitung 24.07.2007

·         A Twist on the Möbius Band. Math in the Media, August 2007

·         Articles about modeling the shape of a Möbius strip. Math Digest, July 2007

·         Möbius at rest. +Plus Magazine 03.08.2007

·         Möbius Problem Solved. 16 July 2007

·         Scoperto il segreto del nastro di Moebius. Newton 27 luglio 2007

·         Le equazioni del nastro di Möbius. Le Scienze 17 luglio 2007

·         La forme du ruban de Möbius n’a plus de secret. Science & Vie No. 1080 Septembre 2007, p. 14

·         Comment se forme un ruban de Möbius ? La Recherche No. 412 Octobre 2007, p. 28

·         Research Highlights 2007: Mind over Möbius. Nature 450, p. 1132,  20 Dec 2007

See also:

 L.Frazier, D.Schattschneider. Möbius bands of wood and alabaster. Journal of Mathematics and the Arts 2(3) 107-122 (2008). DOI: 10.1080/17513470802222926

H.Dambeck. Rätsel des Möbiusbands gelöst. In: Numerator. Mathematik für jeden, S. 132-136. ISBN: 344215572X





Figure: Computed Möbius strip of aspect ratios 5π, 2π, π, 2π/3.

Colour shows the bending energy density (from violet for almost flat regions to red for highly bent).