The topology of the torus

The torus is officially
T =S1 x S1
={(u,v,x,y) in R4 | u2 + v2 = x2 + y2 = 1}
={(cos(q), sin(q), cos(f), sin(f)) | q,f in R}
This picture illustrates the map f:T->R3 given by
f(u,v,x,y) = (2u,2v,0) + x(u,v,0) + y(0,0,1)
=((2+x)u, (2+x)v, y)
The point P in the diagram is
f(cos(q), sin(q), cos(f), sin(f)),
where q is the angle shown in yellow, and f is the angle shown in green.

This animation shows how the torus can be constructed by gluing together opposite edges of a square.

This animation shows that the torus with one point removed is homotopy equivalent to a figure eight.

If we collapse the figure eight down to a point, then the resulting space is homemorphic to the sphere S2

This animation shows that the loop aba-1b-1 is nullhomotopic, illustrating the fact that the fundamental group of the torus is abelian.