# Möbius strip

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

We can parameterize the Möbius strip by

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

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

Topologically, the Möbius strip is formed by taking a quotient space^{} of ${I}^{2}=[0,1]\times [0,1]\subset {\mathbb{R}}^{2}$. 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.

Title | Möbius strip |
---|---|

Canonical name | MobiusStrip |

Date of creation | 2013-03-22 12:55:28 |

Last modified on | 2013-03-22 12:55:28 |

Owner | Mathprof (13753) |

Last modified by | Mathprof (13753) |

Numerical id | 22 |

Author | Mathprof (13753) |

Entry type | Definition |

Classification | msc 54B15 |

Synonym | Möbius band |

Related topic | KleinBottle |

Related topic | Torus |