Main Concept

The Möbius strip is a simple example of a one-sided, non-orientable surface. It is very simple to construct: take a rectangular piece of paper, give it a half-twist (by 180°) (or, an odd number of such twists) and then join the ends together. If you trace a line with your pencil along the strip, you will reach the starting point, but on the other side of the paper, which is strange because you have not ever crossed over the edge! As well as having only one side, the Möbius strip also has only one edge.

The Möbius strip is called a non-orientable surface because you cannot define a continuously varying unit normal vector at each point on the surface. By contrast, a sphere is an orientable surface since you can define on it a normal vector that is continuous, for example, by defining it to point into the interior of the sphere. Möbius strips are also chiral, which means that they either twist clockwise or counter-clockwise as one moves along the surface. A clockwise-twisting Möbius strip and a counter-clockwise-twisting one cannot be made to fit together.

Parameterization

A parameterization of the Möbius strip in ${\mathrm{\ℝ}}^3$ is:   $x\left(\mathrm{\θ}\,w\right)\= \left(R\+\frac{w}{2}\cos \frac{n \mathrm{\theta }}{2}\right) \cos \mathrm{\θ}\,$ $$ $y\left(\mathrm{\theta }\,w\right)\= \left(R\+\frac{w}{2}\cos \frac{n \mathrm{\theta }}{2}\right) \sin \mathrm{\θ}\,$   $z\left(\mathrm{\θ}\,w\right)\=\frac{w}{2} \sin \frac{n \mathrm{\θ}}{2}\,$ where $R$ is the radius of the center circle of the strip, $w$ is the width, $n$ is the number of half-twists, and $\mathrm{\θ}$ is a parameter which runs from $0$ to $2 \mathrm{\π}$. If $n\=1$, you obtain the classic Möbius strip, but for large $n$ you get a 'more twisted' shape. For even $n$ the surface is orientable and has a well-defined principal normal, while for odd $n$ it is non-orientable.$$

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