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 obius band (or 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 obius strip cannot
be represented as the solution set of a ‘nice’ equation in the sense
discussed above.
A related counterintuitive property of the obius strip has to do
with closed curves. In the plane, any closed curve divides the plane
into two
regions2—on
the 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!
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