M\"obius strip MobiusStrip 2002-08-08 03:02:41 2006-10-02 13:40:35 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) \usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here A \emph{M\"{o}bius 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 \PMlinkescapetext{side}. We can parameterize the M\"{o}bius strip by \[ x = r \cdot \cos{\theta}, \quad y = r \cdot \sin{\theta}, \quad z = (r-2)\tan{\frac{\theta}{2}}. \] The M\"{o}bius strip is therefore a subset of the solid torus. Topologically, the M\"{o}bius 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 M\"{o}bius strip. Schematically we can represent this identification as follows: \begin{center} \includegraphics[scale=0.5]{mobius-2} \\ \tiny{Diagram 1: The identifications made on $I^2$ to make a M\"{o}bius strip. \\ We identify two opposite sides but with different orientations.} \end{center} Since the M\"{o}bius 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.