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.)
Created: July 8, 1995 ---
Last modified: Jun 14 1996
Copyright © 1995-1996 by
The Geometry Center
All rights reserved.