TITLE: What makes Torus Special
QUESTION [5 upvotes]: For the past couple of days I have been encountering the word "Torus" quite often. 

I would like to know what special properties does the Torus possess that it is studied very much in Mathematics. 

A recent article on Toral Automorphisms was given out to students by one of our Professors, which i have posted here.

http://chandrumath.wordpress.com/2011/01/31/toral-automorphisms/

This is one example which illustrates what properties "Torus" possesses. I am looking for more exciting properties of the Torus which makes it ubiquitous in Mathematics. Another question :

Are Torus and Doughnuts both same? Or is there any topological difference between them.

REPLY [4 votes]: One of your questions was "Is a torus a doughnut?" The 2-dimensional surface that we know as a torus is the 2-dimensional surface of a standard doughnut or bagel. It is also the 2-dimensional surface of a coffee cup up to homeomorphism (read topological equivalence). This is why we say that a topologist cannot tell the difference between a doughnut and a coffee cup.
If a piece of rope (or better extension cord) is tied loosely into a knot and the ends joined together (or plugged into one another, respectively) then the boundary of the resulting figure is represented as torus. Three dimensional compact spaces that have no incompressible tori are important because of the geometrization conjecture. 
To paraphrase Homer (Simpson, not the poet), "MMM. Tori..."