A Pinch of Topology

Topology -- an important area of mathematics that has applications from subatomic physics to large-scale astronomy -- was born when mathematicians realized that there is something behind the notion of nearness that does not depend on measurements. This qualitative view of nearness implies that elastic distortions, compressions or expansions don't change the essence of a shape. By contrast, tearing an object changes it drastically: points on opposite sides of the tear end up apart, no matter how close together they were. Puncturing -- that is, taking away a point -- is also an essential change. So the rule about not tearing or puncturing the surface is a topological one.

Topology deals with all sorts of surfaces (and also curves). A subarea called differential topology concentrates on surfaces that are smooth. Think of a mesh on the surface, like the latitude-longitude grid on the globe. Smoothness means that the curves of the mesh don't have corners. This corresponds to the intuitive idea of a smooth surface, without creases, kinks, corners, etc. At a crease (such as the equator in the figure above) it's impossible to find a smooth mesh.

It is also impossible to find a smooth mesh around a pinch point, like the point at the center of this surface:

(Click on the picture to rotate it. Check the instructions if necessary.)

This surface is called the Whitney umbrella, in honor of Hassler Whitney, one of the creators of differential topology. You can obtain it by taking an X made of two horizontal lines, and dragging it up while at the same time closing the angle between the lines. The center of the X traces the vertical line of self-intersections that ends at the pinch point, where the two lines collapse into one. 1

Any point that cannot be included in a smooth mesh is called a singular point. Singular points are not ``bad'' -- in fact they can be quite interesting -- but our rules don't allow them because we want our surface to remain smooth. Yorick's attempts to turn the sphere inside out by just pushing the halves through each other fail because he's creating singular points: a crease the first time around, and a pinch point the second time.

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