# torus

Visually, the torus looks like a doughnut. Informally, we take a rectangle^{}, identify two edges to form a cylinder^{}, and then identify the two ends of the cylinder to form the torus. Doing this gives us a surface of genus one. It can also be described as the Cartesian product of two circles, that is, ${S}^{1}\times {S}^{1}$. The torus can be parameterized in Cartesian coordinates^{} by:

$$x=\mathrm{cos}(s)\cdot (R+r\cdot \mathrm{cos}(t))$$ |

$$y=\mathrm{sin}(s)\cdot (R+r\cdot \mathrm{cos}(t))$$ |

$$z=r\cdot \mathrm{sin}(t)$$ |

with $R$ the *major radius* and $r$ the *minor radius* are constant, and $s,t\in [0,2\pi )$.

Figure 1: A torus generated with Mathematica 4.1

To create the torus mathematically, we start with the closed subset $X=[0,1]\times [0,1]\subseteq {\mathbb{R}}^{2}$. Let ${X}^{*}$ be the set with elements:

$$ |

$$ |

and also the four-point set

$$\{0\times 0,1\times 0,0\times 1,1\times 1\}.$$ |

This can be schematically represented in the following diagram.

Diagram 1: The identifications made on ${I}^{2}$ to make a torus.

Opposite sides are identified with equal orientations, and the four corners

are identified to one point.

Note that ${X}^{*}$ is a partition^{} of $X$, where we have identified opposite sides of the square together, and all four corners together. We can then form the quotient topology induced by the quotient map $p:X\u27f6{X}^{*}$ by sending each element $x\in X$ to the corresponding element of ${X}^{*}$ containing $x$.

Title | torus |

Canonical name | Torus |

Date of creation | 2013-03-22 12:55:17 |

Last modified on | 2013-03-22 12:55:17 |

Owner | Daume (40) |

Last modified by | Daume (40) |

Numerical id | 15 |

Author | Daume (40) |

Entry type | Definition |

Classification | msc 54B15 |

Classification | msc 51H05 |

Related topic | MobiusStrip |

Related topic | NTorus |

Related topic | SurfaceOfRevolution2 |

Defines | major radius |

Defines | minor radius |