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torus

(Definition)

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:

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

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

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

with

the major radius and

the minor radius are constant, and $s,t \in [0,2\pi)$ .

$\includegraphics[scale=0.8]{torus}$
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 be the set with elements:

$\displaystyle \{ x \times 0, x \times 1 \mid 0 < x < 1 \}$

$\displaystyle \{ 0 \times y, 1 \times y \mid 0 < y < 1 \}$

and also the four-point set

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

This can be schematically represented in the following diagram.

$\includegraphics[scale=0.5]{torus-2}$
Diagram 1: The identifications made on

to make a torus.
Opposite sides are identified with equal orientations, and the four corners
are identified to one point.

Note that is a partition of , 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 $p\colon X \longrightarrow X^*$ by sending each element $x \in X$ to the corresponding element of containing .

"torus" is owned by Daume. [ full author list (2) | owner history (1) ]

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See Also: Möbius strip, $n$

-torus, surface of revolution

Also defines:

major radius, minor radius

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Cross-references: quotient map, induced, quotient topology, square, partition, point, orientations, opposite sides, diagram, closed subset, Mathematica, Cartesian coordinates, circles, Cartesian product, genus, surface, cylinder, edges, rectangle
There are 32 references to this entry.

This is version 12 of torus, born on 2002-08-07, modified 2007-07-01.
Object id is 3274, canonical name is Torus.
Accessed 11740 times total.

Classification:

AMS MSC:	54B15 (General topology :: Basic constructions :: Quotient spaces, decompositions)
	51H05 (Geometry :: Topological geometry :: General theory)

Pending Errata and Addenda

None.[ View all 7 ]

Discussion

forum policy

Remarks on notation and other stuff by NeuRet on 2002-08-07 15:05:58

1) The elements of a Cartesian product X \times Y are ordered pairs, most commonly denoted by (x,y), not x \times y. It is a bit confusing to think about 0 \times 1, for it feels as if you were referring to 0 and 1 as sets (which they are, but only if we're talking about set theory and ordinals). Why not call it (0,1) instead?

2) It is not clear to me how this torus differs from a Klein bottle (but maybe I'm just being purposely dense). The orientation of opposite edges is important. E.g.

+-<-+
| |
^ ^ <---- This is a torus
| |
+-<-+

+-<-+
| |
^ ^ <----- This is a Klein bottle
| |
+->-+

3) If you follow akrowne's advice and get a little more explicit, why not give your torus a definite shape, i.e. a parametrization? I particularly like

(cos(s)*(R + r*cos(t)), sin(s)*(R + r*cos(t)), r*sin(t))

for 0 <= s,t <= 2pi.

[ reply | up ]

Re: Remarks on notation and other stuff by dublisk on 2002-08-08 02:14:47

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