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

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

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

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

Title Möbius strip MobiusStrip 2013-03-22 12:55:28 2013-03-22 12:55:28 Mathprof (13753) Mathprof (13753) 22 Mathprof (13753) Definition msc 54B15 Möbius band KleinBottle Torus