6 1. Various Ways of Representing Surfaces and Examples

Figure 1.5. Two ways of gluing ends together.

and gluing together two opposite edges. How are we to represent such

a surface by an equation?

One possibility is to add an auxiliary inequality—for example,

one particular bounded cylinder is given as the solution set of

x2

+

y2

=

R2, z2

≤ 1.

This method solves the problem in some cases, but not all. Consider

the second description of a cylinder given above, in which we take

a band of paper and glue together the two ends—now look at what

happens if we twist the band halfway around before gluing the ends

together! The result is the famous M¨ obius band (or M¨ obius strip),

shown in Figure 1.5. Its most surprising property is that it only has

one side: an insect which crawls once around the band will find itself

at the same place, but on the opposite side of the surface.

Now any surface which is given by an equation F (x, y, z) = 0

(with or without inequalities) and which does not contain any critical

points must have two sides—the function F is positive on one side

and negative on the other. It follows that the M¨ obius strip cannot

be represented as the solution set of a ‘nice’ equation in the sense

discussed above.

A related counterintuitive property of the M¨ obius strip has to do

with closed curves. In the plane, any closed curve divides the plane

into two

regions2—on

the M¨ obius strip, though, we can draw closed

curves which have no “inside” or “outside”. Consider the curve which

divides the strip in half, so to speak, running halfway between the free

2This

if the Jordan Curve Theorem, which we will state and prove rigorously in

Lectures 34 and 35. It is not as easy as one might first think!