Professor Stewart's Hoard of Mathematical Treasures (40 page)

19.
Analyticity of Solutions in Calculus of Variations

20.
Boundary Value Problems

21.
Existence of Differential Equations with Given Monodromy

22.
Uniformisation using Automorphic Functions

23.
Development of Calculus of Variations

24.
Simplicity in Proof Theory

Remove exactly two matches to leave two equilateral triangles.

 

Take two matches away, and leave two triangles.

Statisticians know that strange things happen when you combine data. One of them is Simpson’s paradox, which I will illustrate with an example.

The Ministry of Health was collecting data on the success of surgical operations. Two hospitals - Saint Ambrose’s Infirmary and Bumbledown General - were in the same area, and the ministry was going to close the less successful of the two.

To the minister, the situation was perfectly obvious: Bumbledown General had a lower death rate, so he would close Saint Ambrose’s Infirmary.

Naturally, the Chief Executive of Saint Ambrose’s Infirmary protested. But he explained that there was a good reason for reconsidering, and asked the minister to break down the figures into two categories: male and female. The minister was reluctant to do so, on the grounds that it was obvious that Bumbledown General would still do better overall. However, it was easier to look at the new data than to argue, so he obtained the corresponding figures, classified by sex.

Note that the numbers add up correctly, to give the original data.

Strangely, Bumbledown General had a worse death rate than Saint Ambrose’s Infirmary in both categories. Yet, when the
figures were combined, Saint Ambrose’s Infirmary had a worse death rate than Bumbledown General.

In the end, the minister had to keep both hospitals open, unable to justify a decision either way if it were to be contested in court.

In 1958, the distinguished American mathematician Stephen Smale, then a postgraduate student, solved an important problem in topology. But his theorem was so surprising that at first his thesis adviser Arnold Shapiro didn’t believe it, pointing out that there was an obvious counterexample. That is, an example that proves the theorem false. One consequence of Smale’s claimed result was that you can turn a sphere inside out using only continuous, indeed smooth, deformations. That is, you can’t tear it, or cut holes in it, and you can’t even make a sharp crease in it.

Intuitively, this seemed absurd. But intuition was wrong, and Smale was right.

Now, we all know that no matter how we twist and turn a balloon, the outside stays on the outside and the inside stays on the inside. Smale’s work does not contradict this, because it permits one type of deformation that you can’t do with a balloon. Namely, the surface is allowed to move through itself. However, it must do so in a smooth way, without creating a sharp crease. If creases are allowed, ‘eversion’ of the sphere, as it is called, is easy. Just push opposite hemispheres through each other, leaving a tube round the equator, and keep pushing so that the tube shrinks and disappears. However, this method creates an ever-sharper crease round the equator, and the technical definitions in Smale’s theorem rule this out.

This is allowed . . .

. . . but this isn’t.

So Smale was right, and the proof of his theorem could in principle be followed step by step to find an explicit method for everting a sphere. However, in practice this was too complicated, and for several years no specific method was known. The first method was devised by Shapiro and Anthony Phillips, and it was the first of what are now called halfway models.

Topologists have long known that some surfaces are ‘onesided’. The best known example is the Möbius band (Cabinet, page 111), and another is the Klein bottle (page 181). A sphere is two-sided: you can paint the inside surface red and the outside blue, say. But if you try to do that with a Möbius band or a Klein bottle, the red paint eventually runs into the blue paint: the apparent ‘inside’ and ‘outside’ surfaces in any small region connect together further round the band.

Now, there is another one-sided surface, the projective plane, which is closely related to a sphere. In fact, you can construct it mathematically by taking a sphere and pretending that diametrically opposite points are the same - in effect ‘gluing’ them together. The resulting surface can’t be represented in three-dimensional space without passing through itself. But it can be ‘immersed’ in three-dimensional space, meaning that parts of it are permitted to pass smoothly through other parts.

Because the projective plane is a sphere with opposite points glued together, it can be pulled apart into a sphere by ungluing the pairs of points, which creates two separate layers, very close together. One of these is in effect the inside of the sphere, the other the outside. However, because the projective plane doesn’t have an inside and outside, it can be pulled apart in two different ways. If we call the layers ‘red’ and ‘blue’, then the colours match
up as the layers are pulled apart in the two different ways, but the red layer is inside for one way and outside for the other, while the blue layer is outside for one way and inside for the other.

How red and blue layers interchange positions at the halfway stage.

The idea for a specific eversion, then, starts in the middle with an immersed projective plane. Pull it apart one way to create a sphere, with red on the outside and blue on the inside. Then deform that sphere, smoothly, until it looks like a normal round sphere, with only its red surface showing. This may not be easy, and it is not even obvious that it can be done, until you try. However, it works.

Now go back to the halfway stage, and pull the projective plane apart the other way, to create a sphere with blue on the outside and red on the inside. Then deform that sphere, smoothly, until it looks like a normal round sphere, with only its blue surface showing.

Fit these two deformations together by running the first one backwards. Now a sphere that is red on the outside and blue on the inside gets scrunged around, smoothly, until opposite pairs of points coincide at the midway projective plane. Pass the layers through each other, and pull them apart according to the second deformation. The result is a sphere that is blue on the outside and red on the inside.