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Möbius strip

Object type: Surface

Definition

The Möbius strip is the surface obtained as two sides of a rectangular sheet (such as a piece of paper) are joined together in the 'opposite direction'. [If they had been joined in the 'same direction', an elliptic cylinder would be the result.] Below a Möbius strip made from a piece of paper is shown, together with a computer-made 3D surface of the same shape.

The most intriguing property of the Möbius strip is that it is a surface with only one side.

A Möbius strip

A Möbius strip

Parameterisation

The Möbius strip shown above is the image $\mathbf{r}(\left[0,2\pi\right[\times\left[-1,1 \right])$ where $$\mathbf{r}(u,v) = \basis\begin{pmatrix}\left(1+\frac{1}{2}v\cos{\frac{u}{2}} \right)\cos u\\\left(1+\frac{1}{2}v\cos{\frac{u}{2}}\right)\sin u\\\frac{1}{2}v\sin{\frac{u}{2} }\end{pmatrix}.$$

Properties

Standard unit normal

Since the Möbius strip is nonorientable, it is not possible to construct a smooth, non-vanishing normal vector field on the entire surface. Nevertheless, one can imagine that one 'cuts open' the strip somewhere along it, so that a (deformed) rectangular piece of paper is obtained. Of course, this paper has two sides, is orientable, and has such a normal vector field. The image below shows such a field, where the Möbius strip is 'cut open' at its front (bottom centre).

A 'normal vector field' of a
Möbius strip

Obviously, one cannot talk about the 'standard unit normal vector field' of Möbius strip. Nevertheless, at each point, there is a well-defined normal direction, and so one may talk about the 'standard normal direction field' of the surface. This field is shown below.

A 'normal direction field' of a
Möbius strip