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:

$$x=\cos (s)\cdot (R+r\cdot \cos (t))$$

$$y=\sin (s)\cdot (R+r\cdot \cos (t))$$

$$z=r\cdot \sin (t)$$

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 $X=[0,1]\times [0,1]\subseteq {ℝ}^2$. 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 $p:X⟶X^{*}$ 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