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 2 This 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!

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2008 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.