Möbius strip

A Möbius strip is a non-orientiable 2-dimensional surface with a 1-dimensional boundary. It can be embedded in ${ℝ}^3$, but only has a single .

We can parameterize the Möbius strip by

$$x=r\cdot \cos \theta ,y=r\cdot \sin \theta ,z=(r-2)\tan \frac{\theta }{2}.$$

The Möbius strip is therefore a subset of the solid torus.

Topologically, the Möbius strip is formed by taking a quotient space of $I^2=[0,1]\times [0,1]\subset {ℝ}^2$. We do this by first letting $M$ be the partition of $I^2$ formed by the equivalence relation:

$$(1,x)\sim (0,1-x)\mathit{\hspace{1em}}\text{where}\mathit{\hspace{1em}}0\le x\le 1,$$

and every other point in $I^2$ is only related to itself.

By giving $M$ the quotient topology given by the quotient map $p:I^2\to M$ we obtain the Möbius strip.

Schematically we can represent this identification as follows:

Diagram 1: The identifications made on $I^2$ to make a Möbius strip.

We identify two opposite sides but with different orientations.

Since the Möbius strip is homotopy equivalent to a circle, it has $\mathbb{Z}$ as its fundamental group. It is not however, homeomorphic to the circle, although its boundary is.