About the shape of the Möbius ribbon

About the shape of the Möbius ribbon

A Möbius tape is an incontrollable surface that has a side and an edge when placed in a 3-D space. The tape was discovered in the first half of the 19th century by two German mathematicians independently of each other. It is named after one of its explorers, August Ferdinand Möbius.

The Möbius strip occurs in a surprising number of areas of life. Many works of art have the exact form (e.g. M. C. Some of Esher’s images), in many cases, symbolize continuity, return to himself. It is also found in nature (for example, in the shape of molecules of aromatic compounds). The Möbius strip can also be discovered in everyday life: for example, in conveyor belts, this shape ensures that the surface wears evenly. In light of all this, it is surprising that the shape of the tape cannot be described by closed mathematical relationships.

When parameterizing a surface in a 3D space, such as a sphere, a clear relationship is established between a plane shape (such as a rectangle) and the points of the surface. In this example, the points in the rectangle correspond to the longitude and width. It is known that these two data clearly identify any point in the sphere, with the exception of the two poles, where the longitude figure is not clear. Maps are also drawn up on the basis of this mapping.

At the moment, there is no known similar mapping of the Möbius strip that can be placed in the plane, despite numerous researches on the specifics of the tape. The Möbius strip can take a number of shapes under different loads, but the above-mentioned studies are mainly aimed at determining the (unladen) equilibrium shape of the tape. The difficulty of this is that tensions (even when unladen) are awakened at the points of the Möbius strip (similar to stretched brackets), which ensure curvature in the shape through balanced torques.

In my research, I am looking for a (approximate) formula that provides a simple mapping between the points of the rectangle and the unladen Möbius tape made of it. This makes the deformations of the Möbius strip due to different loads easier to calculate than in today’s procedures.

V.B. Gerritsen: The protein with a topological twist (2002) – https://web.expasy.org/spotlight/back_issues/020/

E.LStarostin, G.H.M van der Heijden. The shape of a Möbius strip. Nat. Mater.6, 563-567 (2007)

S.Bartels,· P.Hornung:BendingPaper and the Möbius Strip. J Elast119:113 (2015)