Just as the rules of baseball have changed over time for balance, and just as you can't write a courtroom thriller based on a case where all the evidence is on one side and the conclusion is foregone, so it is in mathematics: the most interesting situations are those where it's not obvious whether or not something can be done---or where it turns out that a solution exists where none was expected.
If everything is allowed -- if you can make holes, for instance -- it's obvious that you can ``turn the sphere inside out'', and Outside In would be a very short movie. The question becomes more interesting when we limit the allowed transformations -- but not too much. That's where the rules come in.
It's best if the rules are ``natural'', not far-fetched or contrived. To say that the surface shouldn't be torn or punctured or creased is natural in the context of differential topology -- a tear or a crease changes the character of the surface more drastically than a mere deformation from roundness.
On the other hand, self-intersecting surfaces may not seem natural at first glance. However, by ``natural'' I don't mean occurring in nature, but rather reasonable from the point of view of analogy or logic. Self-intersecting surfaces are natural by analogy with curves, and also because one way that surfaces are defined in mathematics is as the set of points in space satisfying a certain equality -- for instance, the sphere is the set of points whose distance to the center is equal to a fixed number. Surfaces formulated in this way often turn out to be self-intersecting, like the one below, which has the property that for every point on the surface the product of the distances to the two parallel lines shown is the same.
(Click on the picture to rotate it. Check the instructions if necessary.)
Comments to: webmaster@www.geom.uiuc.edu Created: July 8, 1995 --- Last modified: Jun 14 1996 Copyright © 1995-1996 by The Geometry Center All rights reserved.