1.1. Behold, the Torus!
The Euclidean plane, denoted
is probably the simplest of all
consists of all points X = (x1,x2) where x1 and x2 are
real numbers. One may similarly define Euclidean 3-space
though the Euclidean plane is very simple, it has the complicating
feature that you cannot really see it all at once: it is unbounded.
Perhaps the next simplest surface is the unit sphere. Anyone who
has played ball or blown a bubble knows what a sphere is. One way
to define the sphere mathematically is to say that it is the solution
to the equation
The sphere is bounded and one can, so to speak, comprehend it all
at once. However, one complicating feature of the sphere is that it
is fundamentally curved. Also, its most basic definition involves a
higher-dimensional space, namely
The square torus is a kind of compromise between the plane and
the sphere. It is a surface that is bounded like the sphere yet flat
like the plane. The square torus is obtained by gluing together the
opposite sides of a square, in the manner shown in Figure 1.1.