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# 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 side.

We can parameterize the Möbius strip by

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

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:

$(1,x)\sim(0,1-x)\quad\mbox{where}\quad 0\leq x\leq 1,$ |

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.

## Mathematics Subject Classification

54B15*no label found*

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## Attached Articles

## Corrections

missing word by Mathprof ✓

Mobius strip by juanman ✓

punctuation by mps ✓

(1,x) ~ (0,1-x)where by alozano ✓