Different ways of thinking about Tori

From: Stephen Lavelle (analytic_at_gmail.com)
Date: 09/29/04 

Date: 29 Sep 2004 04:08:15 -0700

I was (idly) thinking about all the different ways you can think about
a sphere, but then I realised that there really wasn't much to that
picture*, so I
started thinking about different ways that you can think of a torus.

Here's what I came up with (i'll keep it brief because it's a maths
forum; for more detail see http://www.maths.tcd.ie/~icecube/tori.html
):

1. Geometric Shape:
A doughnut or a ring or whatever you want to call it as encountered in
everyday life.

2. Equation:
        [c-sqrt(x^2+y^2)]^2+z^2=a^2

3. Quotient Topology:
[0,1]x[0,1] with the equivalences [x,0]=[x,1]
and [0,y]=[1,y]. This gives the view of a torus as a quotient space,
and also as a plane-tiling.

4. Fibre Bundle:
Take a circle, and to any point on that circle associate another
circle.

5. Infinite Cyclic Group:
Fundamental group of a torus. Thinking in terms of holes.
   
6. Riemann Surface:
The function
        w(z)=sqrt[(z^2-1)(z^2-2)]
Has a surface that can be nicely represented on a torus.
   
Anyone have any other ideas? Or can anyone think of anything
interesting for other shapes that have significantly different
properties**? (I was going to include the idea of it as a surface of
revolution, and the symmetries that the quotient map gives, but
they're not really fundamentally different viewpoints).

All in the name of light-hearted fun...

Stephen

*this is probably out of ignorance I know; I would be delighted if
someone could tell me some cool things about spherical thingies.

**point 6 can work also as interpreting the complex multifunction as a
torus, but I don't know enough complex analysis to give many other
representations

Relevant Pages

  • Re: easiest and shortest proof of Jordan Curve theorem
    ... I think you need to work on your spatial intuition. ... I know I failed with that definition of given two points make it act as a radius to sweep out an interior of a circle on the torus. ...
    (sci.math)
  • Re: easiest and shortest proof of Jordan Curve theorem
    ... full interior to that circle, ie, all points of the circle and its ... interior are points on the surface of the torus. ... line on a sphere may look like a line of the sphere but is not a line ...
    (sci.math)
  • Re: easiest and shortest proof of Jordan Curve theorem
    ... > Some circles on the torus have interiors whose points all lie on the ... > surface of the torus. ... > the torus whose interior are points that lie outside the torus itself. ... Fine, but given a circle on the torus, or any other surface, how do you ...
    (sci.math)
  • Re: easiest and shortest proof of Jordan Curve theorem
    ... > On a torus, the JCT is false. ... > simple Euclidean plane geometry, but the JCT applies to topological ... but rather the circle in that given any two points a unique circle is ... those circles formed on a perpendicular to the hole would no longer be ...
    (sci.math)
  • Re: easiest and shortest proof of Jordan Curve theorem
    ... full interior to that circle, ie, all points of the circle and its interior are points on the surface of the torus. ... There are many such circles with their interiors on the torus surface. ... So if the above is well defined, then obviously a torus obeys the JCT ...
    (sci.math)