Chapter 1

Book Overview

1.1. Behold, the Torus!

The Euclidean plane, denoted

R2,

is probably the simplest of all

surfaces.

R2

consists of all points X = (x1,x2) where x1 and x2 are

real numbers. One may similarly define Euclidean 3-space

R3.

Even

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

set, in

R3,

to the equation

x1

2

+ x2

2

+ x3

2

= 1.

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

R3.

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.

1

http://dx.doi.org/10.1090/stml/060/01