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, . The torus can be parameterized in Cartesian coordinates by:
Figure 1: A torus generated with Mathematica 4.1
and also the four-point set
This can be schematically represented in the following diagram.
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 by sending each element to the corresponding element of containing .
|Date of creation||2013-03-22 12:55:17|
|Last modified on||2013-03-22 12:55:17|
|Last modified by||Daume (40)|